Am I Getting a Good Deal? Reference-Dependent Decision Making when the Reference
Price is Uncertain
Jayson L. Lusk, Vincenzina Caputo, and Rodolfo M. Nayga, Jr.*
Abstract: Models of consumer decision making commonly incorporate reference-dependent
preferences. In these models, the reference point is typically assumed to be known to the
consumer. However, research on price recall and the reference formation process reveals that for
most consumers the reference price is often uncertain at the time of purchase. Reference price
uncertainty, which arises from price variability, creates ambiguity for the consumer about
whether they are getting a good or bad deal. This paper develops a model of consumer choice in
the presence of reference-dependent preferences when the reference price is uncertain. Data
from three separate studies that vary the product category, use of brand names, and the way
reference prices and reference price uncertainty are measured show that the new model generally
better describes consumer choice than conventional discrete choice models that either ignore
reference prices or treat reference prices as certain. Incorporating reference price uncertainty
tends to yield own-price elasticities that are more elastic (sometimes twice as elastic) than those
from more conventional reference-dependent models that treat the reference price as certain.
The new model provides insights into the effect of reducing reference price uncertainty on
consumer choice.
Keywords: reference-dependent preferences; reference price uncertainty; consumer demand;
discrete choice models
JEL codes: D03, D11, C25, Q11
Contact: Jayson Lusk, 411 Ag Hall, Oklahoma State University; Stillwater, OK, 74074;
[email protected]; (405) 744-7465.
*Authors are Regents Professor and Willard Sparks Endowed Chair, Oklahoma State University;
Assistant Professor, Korea University; and Professor and Tyson Endowed Chair, University of
Arkansas and adjunct professor, Korea University and Norwegian Agricultural Economics
Research Institute.
Introduction
Motivated in part by the work of Kahneman and Tversky (1979) and Tversy and Kahneman
(1991), models of reference-dependent preferences have become ubiquitous in the study of
consumer choice. Nowhere is this more true than in the analysis of reference-prices and their
influence on shopping behavior (Briesch et al., 1997; Kalyanaram and Winer 1995; Mazumdar,
Raj, and Sinha, 2005; Neumann and Böckenholt, 2014; Winer 1995). Recent empirical
applications have incorporated reference price effects into models that explain travel choice
(Rose and Masiero, 2010), automobile demand (Mabit and Fosgerau, 2011), selection of travel
destination (Nicolau, 2011), environmental valuations (Hasund et al., 2011), and food choice (Hu
et al., 2006), among others. This literature captures the idea that consumers are loss-averse with
regard to price changes; prices above a reference price signal a “bad deal” (Weaver and
Frederick, 2012), and as such, price sensitivity is more pronounced above a reference price than
below.
Despite the widespread acceptance of reference-dependent models in economics,
psychology, and marketing, typical approaches tend to omit a potentially important behavioral
insight. Consumers are unlikely to know their reference prices with certainty. Introspectively,
many shoppers can attest to having faced uncertainty over whether they are “getting a good
deal”. How does reference price uncertainty affect choice, and what is the consequence of
ignoring this phenomenon when predicting consumer behavior? This paper addresses these
questions by developing and empirically estimating a model of consumer choice in the presence
of reference-dependent preferences when the reference price is uncertain.
That reference-price uncertainty has been largely overlooked is curious given several
strains of literature suggesting that this feature of the decision-making process is likely to be the
1
rule rather than the exception. First, it has commonly been assumed that reference-dependence is
a result of memory-based reference prices or a result of internal reference prices measured as a
function of prices previously paid (Briesch et al. 1997; Mazumdar, Raj, and Sinha, 2005).
However, there is ample evidence suggesting that consumers are unaware of past prices paid for
products – even products they just placed in their grocery cart (Dickson and Sawyer 1990;
Estelami and Lehmann, 2001; Urbany and Dickson 1991; Vanhuele and Drèze 2002; Moon et al.
2006). Studies typically find that less than 50% of shoppers can recall the exact prices paid for
products (Jensen and Grunert, 2014). If prices previously paid are not known with any degree of
accuracy, it is unlikely that reference-prices are known with certainty. Second, the growing
research on the reference-point formation process (e.g., Abeler et al., 2011; Baucells, Weber, and
Welfens, 2011; Ericson, and Fuster, 2011; Kőszegi and Rabin, 2006) highlights the fluidity of
the reference points and emphasizes the role of expectations in determining the reference price.
Given that expectations implicitly imply underlying stochasicity, it follows that reference-points
are unlikely deterministic. Finally, even in models which implicitly assume that reference prices
are known by the consumer, researchers typically use household scanner data to estimate each
consumer’s reference price (e.g., Briesch et al., 1997; Erdem, Mayhew, and Sun, 2001; Bruno,
Che, and Dutta, 2012). The econometrically-estimated reference prices are thus projected with
some level of uncertainty, which is typically ignored by analysts when fitting subsequent choice
models.
Our inquiry bears some similarity with many strains of related literature. For example,
some research has suggested that consumers have thresholds of price acceptance, and thus appear
to utilize something like reference-price bounds, rather than a single reference price (e.g., Han,
Gupta, and Lehmann, 2002; Terui and Dahana, 2006). Ghoshal et al. (2014) argued that
2
consumers encounter communications and brands sequentially, and a variety of prior experiences
serve to create multiple reference points. There is also a rich body of research on the extent to
which consumers undertake search costs to better form expectations to improve subsequent
choice (e.g., De los Santos, Hortacsu and Wildenbeest, 2012; Kim, Albuquerque, and
Bronnenberg, 2010; Mehta, Rajiv, Srinivasan, 2003), findings which relate to the argument that
reference price uncertainty arises from a lack of price knowledge. Related to search costs is the
concept of option-values or “commitment costs”, which arise when consumers have the ability to
delay a purchase and are uncertain about their ability to acquire information about the good in
the future (Zhao and Kling, 2004). As discussed in the reviews of Briesch et al. (1997) and
Mazumdar, Raj, and Sinha (2005), there are a number of studies analyzing whether reference
price(s) are internal (e.g., based on expectations or memories of prices previously paid) or
external (e.g. based on external notions of fairness or willingness-to-pay). The model developed
in this paper bears some superficial similarity to the general conceptual model derived by
Kőszegi and Rabin (2006), who consider a model with uncertainty in prices, where the reference
point is stochastic and endogenously determined.
Previous research, thus, recognizes the fact that consumers face price variability (as
indicated by the search cost literature), may have multiple reference points (as indicated by the
literature on price acceptance thresholds), and may interpret price changes or variability as
quality signals which may also affect reference points (Hardie, Johnson, and Fader, 1993). It is
not likely that any single model will unify all these strands of literature in a way that well
describes all contexts, nor is it likely that all the previously discussed issues are separately
identifiable (i.e., if there is absolutely no price variability, there is no need to search, and it is
unlikely that internal reference prices would exist). To the extent that past prices inform current
3
reference points, it seems reasonable to believe that previous (or currently expected) price
variability is intrinsically linked to reference-price variability (or what we call reference price
uncertainty).
The particular situation we aim to model is akin to shopping for a grocery item in a
particular store. In such cases, prices for each brand at a given retailer (and at the time of choice)
are known with certainty. Because the costs of the individual items are typically small relative to
income, consumers are unlikely to (at least at the time of item choice) engage in price search
across retailers. Thus, there is no uncertainty in the prices being charged for each item at the
given retailer (i.e., the prices are within eyesight), and at least for known brands, it is unlikely the
case that there is much of a price-quality correlation (and in our empirical analysis, prices are
uncorrelated with brands). However, there might be uncertainty about prices paid in the past or
charged by other retailers. In this context, most prior research has utilized consumers’ expected
price (or the price previously paid for the item) before they walked in the door to the retailer as
the reference point. Our model expands on this idea and considers not only what price a
consumer expects to be charged for an item but also about the expected variability of prices they
would expect to be charged across retailers. Just as much of previous work has used the mean
price expectation as the reference point, we use people’s stated level of expected price variability
in their market as the measure of reference price uncertainty.
The next section of the paper presents the model of reference-dependent preferences in
the presence of reference-price uncertainty. The following section discusses a few issues related
to the econometric estimation as a way of transitioning to the empirical analysis. The following
three sections present three separate empirical studies where the new model is implemented and
compared to conventional models that either ignore reference prices or assume reference-price
4
certainty. Three separate studies are conducted to determine the generalizability of our findings
and to determine how robust the results are with regard to: (i) variation in the product category,
(ii) use of brand names, and (iii) the way reference prices and reference price uncertainty are
measured.
All three studies are based on choice-experiments (Louviere, Hensher, and Swait, 2000)
conducted among random samples of the U.S. population. We test our model in three different
studies in three different product categories given that purchase frequencies and price knowledge
likely varies across product categories. Study 1 involves choices between four infrequentlybought branded products (ketchup), where the mean and standard deviation of reference prices
are elicited by asking respondents to state prices associated with 90% confidence intervals.
Study 2 involves choices between frequently-bought fresh milk options that vary in fat content.
The final study investigates choices between six un-branded fresh meat items and two unbranded, non-meat items selected from a grocery store for a meal; reference prices are elicited
for each product as is the uncertainty in each reference-price on a simple slider scale.
Adamowicz and Swait (2012) report the wide variation in repeat same-brand purchases for the
products utilized in this study; 55% of consumers’ milk purchases are of the same previously
purchased brand whereas the only 22% of ketchup purchases meet this criterion. Most fresh beef
and pork sold in grocery stores is unbranded. This variation in product category familiarity
provides variation in reference price uncertainty to test the robustness of the model. The final
section summarizes the findings and concludes with thoughts for future research.
Conceptual Model
5
Assume that individual i evaluates the desirability of a purchase option based on the product’s
price, 𝑝𝑗 , relative to some internal, subjective price reference point, 𝑟𝑖𝑗 , for the product.
Consumers perceive prices above the reference point (losses) differently than changes below the
reference point (gains), such that an individual’s utility from purchasing option j is:
(1)
𝑈𝑖𝑗 = 𝑉𝑖𝑗 + 𝜀𝑖𝑗 = 𝛼𝑗 (𝑝𝑗 − 𝑟̃𝑖𝑗 )𝐼𝑟𝑖𝑗<𝑝𝑗 + 𝛽𝑗 (𝑝𝑗 − 𝑟̃𝑖𝑗 )𝐼𝑟𝑖𝑗>𝑝𝑗 + 𝜏𝑗 + 𝜀𝑖𝑗
where 𝐼𝑟𝑖𝑗<𝑝𝑗 is an indicator variable taking the value of one when prices are higher than the
reference point (losses), 𝐼𝑟𝑖𝑗>𝑝𝑗 is an indicator variable taking the value of one when prices are
lower than the reference point (gains), and 𝛼𝑗 and 𝛽𝑗 are price-sensitivity parameters, where it is
assumed that 𝛼𝑗 < 𝛽𝑗 , implying that consumers are loss averse. The parameter 𝜏𝑗 represents an
alternative-specific constant indicating the utility of option j that is unexplained by price
variation, and 𝜀𝑖𝑗 reflects an idiosyncratic effect unobserved to the analyst.
Variations on the specification in (1) above have been explored in detail in previous
demand studies, and there is ample evidence to suggest support for the reference-price model,
i.e., that 𝛼 ≠ 𝛽 (Mazumdar, Raj, and Sinha, 2005). The key departure we take from these
previous studies is the intuitive assumption that individuals are uncertain about the reference
price and that, from the perspective of the individual, 𝑟𝑖𝑗 is a subjective random variable, as
denoted by 𝑟̃𝑖𝑗 in equation (1). 𝑟̃𝑖𝑗 is assumed to be distributed according to the probability
density function, 𝑓𝑖 (𝑟𝑖𝑗 ) and the cumulative distribution function 𝐹𝑖 (𝑟𝑖𝑗 ).
They key question is how individuals evaluate the desirability of a choice option when
the reference price is uncertain (i.e., when consumers are unsure of whether they are getting a
“good deal”). One approach is to assume consumers are risk neutral and that they evaluate the
6
desirability of an option based on the expected utility of the option, as given by the following
(the i subscript has been dropped for notational convenience):
(2)
𝑝
∞
𝐸[𝑉𝑗 ] = ∫−∞ 𝛼𝑗 (𝑝𝑗 − 𝑟𝑗 ) 𝑓(𝑟𝑗 )𝑑𝑟𝑗 + ∫𝑝 𝛽𝑗 (𝑝𝑗 − 𝑟𝑗 ) 𝑓(𝑟𝑗 )𝑑𝑟𝑗 + 𝜏𝑗 .
The first integral is taken over all potential reference points less than the product price, in which
case the shopper experiences a loss and evaluates price changes according to the parameter 𝛼𝑗 .
The second integral is taken over all potential reference points greater than the product price, in
which case the shopper experiences a gain and evaluates price changes according to the
parameter 𝛽𝑗 . Expanding equation (2) yields:
(3)
𝑝
𝑝
∞
∞
𝐸[𝑉𝑗 ] = 𝛼𝑗 𝑝𝑗 ∫−∞ 𝑓(𝑟𝑗 )𝑑𝑟𝑗 − 𝛼𝑗 ∫−∞ 𝑟𝑗 𝑓(𝑟𝑗 )𝑑𝑟𝑗 + 𝛽𝑗 𝑝𝑗 ∫𝑝 𝑓(𝑟𝑗 )𝑑𝑟𝑗 − 𝛽𝑗 ∫𝑝 𝑟𝑗 𝑓(𝑟𝑗 )𝑑𝑟𝑗 +
𝜏𝑗
𝑝
Note that ∫−∞ 𝑓(𝑟𝑗 )𝑑𝑟𝑗 = 𝐹(𝑝𝑗 ) is the probability that the reference point is less than the
∞
observed price and that ∫𝑝 𝑓(𝑟𝑗 )𝑑𝑟𝑗 = 1 − 𝐹(𝑝𝑗 ) is the probability that the reference point is
greater than the observed price. If the subjective reference price distribution is Normally
distributed then, 𝐹(𝑝𝑗 ) = 𝛷((𝑝𝑗 − 𝑟̅𝑗 )/𝜎𝑗 ) where 𝑟̅𝑗 is the mean subjective reference price (from
the particular consumer’s perspective), 𝜎𝑗 is the standard deviation of the subjective reference
price (again, from the consumer’s perspective), and 𝛷 is the standard Normal cdf. Substituting,
we how have:
(4)
𝑝
∞
𝐸[𝑉𝑗 ] = 𝛼𝑗 𝑝𝑗 𝐹(𝑝𝑗 ) − 𝛼𝑗 ∫−∞ 𝑟𝑗 𝑓(𝑟𝑗 )𝑑𝑟𝑗 + 𝛽𝑗 𝑝𝑗 (1 − 𝐹(𝑝𝑗 )) − 𝛽𝑗 ∫𝑝 𝑟𝑗 𝑓(𝑟𝑗 )𝑑𝑟𝑗 + 𝜏𝑗 .
𝑝
The expression ∫−∞ 𝑟𝑗 𝑓(𝑟𝑗 )𝑑𝑟𝑗 is the mean of the truncated distribution for r (truncated
𝑝
from above at p) multiplied by the probability of r being less than p. That is, ∫−∞ 𝑟𝑗 𝑓(𝑟𝑗 )𝑑𝑟𝑗 =
𝐸[𝑟𝑗 |𝑟𝑗 < 𝑝𝑗 ]𝐹(𝑟𝑗 ). For a Normal distribution, 𝐸[𝑟𝑗 |𝑟𝑗 < 𝑝𝑗 ] = 𝑟̅𝑗 − 𝜎𝑗 [𝜙(𝜃)/𝛷(𝜃)], where
7
𝜃 = (𝑝𝑗 − 𝑟̅)/𝜎
𝑗
𝑗 , and where 𝜙 is the standard normal pdf. For the case where truncation is from
∞
below (i.e., when the reference price exceeds observed price), then ∫𝑝 𝑟𝑗 𝑓(𝑟𝑗 )𝑑𝑟𝑗 =
𝐸[𝑟𝑗 |𝑟𝑗 > 𝑝𝑗 ](1 − 𝐹(𝑟𝑗 )) and for a Normal distribution, 𝐸[𝑟𝑗 |𝑟𝑗 > 𝑝𝑗 ] = 𝑟̅𝑗 + 𝜎𝑗 [𝜙(𝜃)/(1 −
𝛷(𝜃))]. Plugging these values into equation (4), and assuming the perceived reference price is
normally distributed, yields:
(5)
𝜙(𝜃)
𝐸[𝑉𝑗 ] = 𝛼𝑗 𝑝𝑗 𝛷(𝜃) − 𝛼𝑗 𝛷(𝜃){𝑟̅𝑗 − 𝜎𝑗 [𝛷(𝜃)]} + 𝛽𝑗 𝑝𝑗 (1 − 𝛷(𝜃)) − 𝛽𝑗 (1 − 𝛷(𝜃)){𝑟̅𝑗 +
𝜙(𝜃)
𝜎𝑗 [1−𝛷(𝜃)]} + 𝜏𝑗 .
Slightly simplifying, equation (5) can be written as:
(6)
𝜙(𝜃)
𝜙(𝜃)
𝐸[𝑉𝑗 ] = 𝛼𝑗 𝛷(𝜃) {𝑝𝑗 − 𝑟̅𝑗 + 𝜎𝑗 [𝛷(𝜃)]} + 𝛽𝑗 (1 − 𝛷(𝜃)){𝑝𝑗 − 𝑟̅𝑗 − 𝜎𝑗 [1−𝛷(𝜃)]} + 𝜏𝑗 .
Equation (6) has an intuitive interpretation. To see this, (6) can be described in general
terms as:
(7)
𝐸[𝑉𝑗 ] = 𝛼𝑗 𝑃𝑟𝑜𝑏[𝑟𝑗 < 𝑝𝑗 ]{𝑝𝑗 − 𝐸[𝑟𝑗 |𝑟𝑗 < 𝑝𝑗 ]} + 𝛽𝑗 𝑃𝑟𝑜𝑏[𝑟𝑗 > 𝑝𝑗 ]{{𝑝𝑗 − 𝐸[𝑟𝑗 |𝑟𝑗 > 𝑝𝑗 ]} +
𝜏𝑗 .
For the first part of the expression, the “loss averse” price parameter, 𝛼𝑗 , is multiplied by the
probability that the observed price exceeds the reference point, 𝑃𝑟𝑜𝑏[𝑟𝑗 < 𝑝𝑗 ], which in turn is
multiplied by the difference between the price of the option, 𝑝𝑗 , and the expected reference price
given that observed price is greater than the reference price (the mean of the reference price
truncated from below at the observed price), 𝐸[𝑟𝑗 |𝑟𝑗 < 𝑝𝑗 ]. The second part of the expression
analogously multiplies the “gain” price parameter, 𝛽𝑗 , by the difference in the observed price and
expected reference point, conditional on the reference point being greater than the price. In this
8
way, (6) and (7) can be seen as a probabilistic expression of the original formulation stated in
(1).
Empirical Model and Estimation
Equation (6) forms the basis for estimation for Studies 1 and 2. In particular, equation (6) is
substituted into the following random utility model
(7)
𝑈𝑖𝑗 = 𝐸[𝑉𝑖𝑗 ] + 𝑒𝑖𝑗 + 𝜀𝑖𝑗 .
Assuming 𝜀𝑖𝑗 are distributed iid type I extreme value, the multinomial logit specification results:
(8)
Prob j is chosen by i =
𝑒
𝐸[𝑉𝑖𝑗 ]+𝑒𝑖𝑗
∑𝐽𝑘=1 𝑒 𝐸[𝑉𝑖𝑘 ]+𝑒𝑖𝑘
.
The term 𝑒𝑖𝑗 reflects a mean-zero, Normally distributed alternative-specific effect.
This specification is a mixed-logit, error-components model. Scarpa, Ferrini, and Willis
(2005) suggest a version of this specification, where 𝑒𝑖𝑗 is the same for all alternatives except the
status-quo (or “no purchase”) option, for which there is no error component. This approach
reflects a parsimonious way to accommodate differential substitution patterns among the statusquo (or “no purchase”) alternative and the other alternatives in the choice set. We specify our
model such that error-component is assumed to be individual-specific to account for the repeated
nature of the choice data. Moreover, the specification reflects the panel nature of our data and
captures some degree of respondent heterogeneity. The utility of the “no purchase” option is
normalized to zero for identification purposes. The likelihood function defined by the
probability statement in (8) does not have a closed form due to the presence of the random term
𝑒𝑖𝑗 . We integrate this term out of the likelihood function using Gaussian quadrature.
To judge the suitability of the new model specification in (6), we estimate four models
that are increasing in complexity. The four models estimated are:
9

Model 1: A standard approach that has single linear price effect that assumes no
reference-dependence (i.e., it is assumed that 𝑉𝑖𝑗 = (𝛼 = 𝛽)𝑝𝑗 + 𝜏𝑗 ).

Model 2: A model with alternative-specific linear price effects that assumes no referencedependence (i.e., it is assumed that 𝑉𝑖𝑗 = (𝛼𝑗 = 𝛽𝑗 )𝑝𝑗 + 𝜏𝑗 ).

Model 3: A reference-dependent model where reference price is assumed known with
certainty (i.e., it is assumed that 𝑉𝑖𝑗 = 𝛼𝑗 (𝑝𝑗 − 𝑟̅𝑖𝑗 )𝐼𝑟̅𝑖𝑗<𝑝𝑗 + 𝛽𝑗 (𝑝𝑗 − 𝑟̅𝑖𝑗 )𝐼𝑟̅𝑖𝑗>𝑝𝑗 + 𝜏𝑗 ).

Model 4: A reference-dependent model with reference price uncertainty (i.e., it is
assumed that 𝑉𝑖𝑗 is defined as in equation (6)).
Models are compared based on the percentage of choices correctly predicted and the AIC
and BIC model selection criteria. Model 1 is a nested version of Model 2, and as such likelihood
ratio tests can be used to test the null that price effects are not alternative specific. Models 3 and
4 do not nest models 1 or 2 via parametric restrictions because of the introduction of the variable
𝑟̅𝑖𝑗 , which is subtracted from 𝑝𝑗 .
Study 1
The first study investigates consumer demand for a relatively infrequently bought item, ketchup,
for which there are well known brand names. Given that ketchup is not bought at a high
frequency (only 11% of our sample report buying ketchup at least once a week) and there are
relatively few repeat-brand purchases, we expected that price may not be as important a factor
driving choice as is the case for other products and reference-prices are likely to be relatively
uncertain.
Methods and Procedures
10
Study 1 utilizes data from 216 subjects, each of whom made 25 choices (for a total of 5,400
choices) between four ketchup brands (i.e., Hunt’s, Heinz, Del Monte, and Red Gold). Each
choice task also included a fifth “no purchase” option that indicated, “if options A, B, C, and D
were all that were available I would not buy ketchup from this store. In each question, the
options were offered at one of five levels ($1.50, $2.25, $3.00, $3.75, and $4.50) chosen to
encompass the range of ketchup prices in major retailer supermarket outlets. Out of the 54=625
different choice questions possible, we selected 25 that comprised an orthogonal fractional
factorial design in which the price of each brand is uncorrelated with all other brands. The choice
questions were randomly ordered across respondents. Data were collected from responses to an
online questionnaire showing images of the ketchup bottles with the respective brands and
prices. Participants were recruited by the online survey software provider Qualtrics and their
associated partners.
After answering the choice questions, respondents’ reference prices were elicited by
asking, “What price would you expect a grocery store to charge for your favorite brand of a 36
ounce (oz) bottle of tomato ketchup?” This approach to eliciting the reference price presumes
that expectations are the appropriate reference point (Ericson, and Fuster, 2011; Kőszegi and
Rabin, 2006) and that the most recently purchased brand’s price is the likely reference (Briesch
et al., 1997). Moreover, in study 1 we only asked about one reference price, and assume it is the
same for all brands – an assumption generally supported by the empirical work of Briesch et al.
(1997). Responses to this question are used to specify the mean price reference point, 𝑟̅𝑖 .
To elicit uncertainty in the reference price, respondents were asked to respond to the
following: “In 90% of stores I would expect to find the price of my favorite brand of a 36 ounce
(oz) bottle of tomato ketchup to be no higher than . . .”. Another similar question was asked
11
except that “higher than” was replaced by “lower than”. To calculate reference price uncertainty
we utilize the answers to these two questions, denoted 𝑟𝑖,0.9 and 𝑟𝑖,0.1, to back out the value 𝜎𝑖
that appears in equation (6). In particular, note that familiar z-score implies that at the 90th
percentile 1.282 = (𝑟𝑖,0.9 − 𝑟̅𝑖 )/𝜎𝑖 , such that 𝜎𝑖 = (𝑟𝑖,0.9 − 𝑟̅𝑖 )/1.282. An analogous value can
be derived using the price stated for the 10th percentile, and we take the mid-point of these two
estimates to define 𝜎𝑖 for each person.
Additional details on the survey questions, along with descriptive statistics are provided
in the appendix.
Estimation Results and Model Comparisons
Table 1 reports the estimates for each of the four models fit to the ketchup data. A likelihood
ratio test rejects model 1 in favor of model 2 (chi-square value of 37 with 3 degrees of freedom;
p-value<0.001), supporting the existence of alternative-specific price effects. All model fit
criteria favor model 3 over model 2, suggesting that the reference-dependent model outperforms
the specifications that ignore consumers’ internal reference prices. Model 4, which incorporates
reference price uncertainty, is the best fitting model. Although it contains the same number of
parameters as model 3, it has a higher percentage of correct predictions and lower AIC and BIC
values.
Discussion
Models 3 and 4 reveal the strong asymmetry that exists between price changes above and below
the reference point. For example, according to model 4, a $1 increase in price for brand 2
12
reduces utility by -2.184 if the price is above the reference point (a loss) but only reduces utility
by -0.482 if the price is below the reference point.
Although model 4 provides a better fit to the data, it is unclear what practical implications
are implied by the model, particularly because the model coefficients appear similar for models 3
and 4. To provide insight into this issue, Table 2 reports the own-price arc-elasticities implied by
each of the models. Because of the potential asymmetry in price effects implied by the
reference-price models, arc-elasticities are calculated for a price increase relative to the midpoint of the prices used in the experimental design ($3) and for a price decrease relative to this
mid-point (note: the mean elicited reference price was $2.62 and the median was $2.5; the mean
implied standard deviation of the subjective reference price distribution was $0.46).
In the absence of the new Model 4 derived in this paper, Model 3 which assumes
reference-dependent preferences but ignores uncertainty in the reference price would be the
preferred specification. As such the comparison of elasticities across models 3 and 4 are most
germane. Model 4 implies substantially more price sensitivity to price increases and decreases
than does model 3. For example, for a price increase, the own-price elasticity of demand for
brand 2 is -1.337 for Model 3 but the same value is -2.468 for Model 4, 1.84 times more elastic.
Somewhat ironically, Model 4 yields own-price elasticities that are, at least for price increases,
more comparable to Models 1 and 2, which completely ignore reference-dependent preferences.
The last rows in table 2 show the effect of reducing the uncertainty in the reference price.
By definition, Models 1, 2, and 3 imply no change in probability of purchase because reference
price uncertainty does not enter into these models. To simulate the effect of such a change for
Model 4, each person’s implied reference price uncertainty,𝜎𝑖 , was reduced by 25%, and the
probabilities of purchase were re-calculated. Table 2 expresses the change in market share in
13
elasticity terms. For example, a 1% reduction in 𝜎𝑖 for each person results in a 0.02% increase in
probability of purchasing brand 2 and a 0.025% reduction in the probability of purchasing brand
3. The effects of a change in reference price uncertainty on probabilities of purchase are small
relative to own-price changes.
Figure 1 also illustrates the practical implications of making inferences based on the
competing models by showing the implied demand curves for Heinz ketchup over the price
ranges utilized in the experimental design (holding prices of all other brands at their means). At
lower price levels (those less than the mean), Model 3 under-predicts market share relative to all
other models, which tend to yield similar market share estimates. At higher prices (higher than
the mean), the new reference price uncertainty model (model 4) yields smaller market shares
than the other models. Over the range of prices likely to be most relevant (say from $2.50 to
$3.50), demand is much more elastic for model 4 than for the other models.
Study 2
The second study moves to a more frequently bought item, milk, for which brand names are less
ubiquitous in a given location and other quality factors – like fat content – are likely to be more
determinative. About 64% of respondents in our sample reported buying milk at least once a
week, and as such, we expected reference price uncertainty to be lower for this product category
than for ketchup.
Methods and Procedures
Study 2 utilizes data from 222 subjects, each of whom made 25 choices (for a total of 5,500
choices) between four fat-content differentiated conventional milk products commonly sold in
grocery stores (skim, 1% reduced fat, 2% reduced fat, and whole) listed at different prices. In
14
addition to these options, each choice task also included a fifth “no purchase” option. Prices of
each fat option varied among five levels ($1.50, $2.25, $3.00, $3.75, or $4.50) across choice
questions. As in study 1, an orthogonal fractional factorial design was generated to reduce the
54=625 possible choice sets to 25 in which the price of each milk option is uncorrelated with the
prices of the other three types of milk. Also, as in Study 1, the order of the choices was
randomly varied across respondents. Data were collected from responses to an online
questionnaire showing images of milk jugs with fat-content labels and prices. Participants were
recruited by the online survey software provider Qualtrics and their associated partners.
Reference prices and reference price uncertainties were elicited in an analogous way to that
described in Study 1. Additional details on the survey questions, along with descriptive statistics
are provided in the appendix.
Estimation Results and Model Comparisons
Table 3 reports the estimates for each of the four models fit to the milk data. A likelihood ratio
test rejects model 1 in favor of model 2 (chi-square value of 43 with 3 degrees of freedom; pvalue<0.001), supporting the existence of alternative-specific price effects. All model fit criteria
favor model 3 over model 2, suggesting that the reference-dependent model outperforms the
specifications that ignore the reference point. As was the case with study 1, Model 4, which
incorporates reference price uncertainty, is the best fitting model according to all three model fit
criteria.
Discussion
15
The estimates in table 3 again reveal a high degree of asymmetry between price changes above
and below the reference point. For example, for model 4 and option 1, the degree of price
sensitivity is 1.490/0.394 = 3.55 times higher for losses relative to “gains”. Table 4 further
illustrates the differences in practical implications implied by each model by reporting own-price
arc-elasticities for price increases and decreases relative to the mid-point of the prices used in the
experimental design ($3.50) (note: the mean elicited reference price was $3.11 and the median
was $3; the mean implied standard deviation of the subjective reference price distribution was
$0.42).
As was the case in study 1, Table 4 shows that for study 2, Model 4 implies substantially
more price sensitivity to price increases and decreases than does model 3. Most notably, for a
price decrease, the implied own-price elasticity of demand for option 1 was -1.753 for Model 3
but was -4.059 for Model 4. Again, Model 4, which includes reference prices and reference
price uncertainty, yields own-price elasticities that are more comparable to Models 1 and 2,
which completely ignore reference-dependent preferences. The last rows in table 4 show the
effect of reducing the uncertainty in the reference price, and again, the effects are relatively
small.
Figure 2 illustrates the demand for whole milk for the four competing models. As was
the case with study 1, the standard reference-dependent model (model 3), under-estimates market
share relative to the other models at lower price levels. By contrast, model 4 suggests lower
market share estimates than the other models at higher price levels. Over the mid-range of
prices, the new model 4 implies more elastic demand that the other models, particularly model 3.
Study 3
16
The third study departs from the first two in several important ways. First, rather than asking
about choices within a narrow product category, respondents were asked to make a main meal
choice among several meat and non-meat items in a grocery setting. Second, rather than
assuming a common reference price for all items, we elicit a reference price for each item.
Finally, because we hypothesized that it might be relatively difficult for respondents to state
reference prices and the 90th and 10th percentiles, the Study 3 utilized an easy-to-answer slider
scale of reference price uncertainty for each item.
Methods and Procedures
This study utilizes data from a choice-experiment conducted among 1,018 respondents, each of
whom made nine choices between six meat and two non-meat items bought in a grocery store for
dinner. Preceding the choice questions was the verbiage: “Imagine you are at the grocery store
buying the ingredients to prepare a meal for you or your household. For each of the following
nine questions that follow, please indicate which meal you would be most likely to buy.”
Each of the questions was identical except that the prices varied across each question.
Each question had nine pictured options (ground beef, steak, pork chop, ham, chicken breast,
chicken wing, rice and beans, pasta, and one “no purchase” option) and the price of each option
was varied at three levels. The prices appearing in each choice were determined by a main
effects orthogonal fractional factorial design. A perfectly orthogonal design (in which prices of
each choice alternative were uncorrelated with each other alternative) required 27 choices. The
27 choices were blocked into three sets of nine, and each person was randomly assigned to one
of the three blocks. The order in which the meal options appeared in the choice set was
randomly varied across subjects.
17
Reference prices for each product were measured by asking respondents about the
expected value of each product’s price in their area. They were asked, “What is your best
estimate of the average price that grocery stores, supermarkets, and wholesale stores charge for
the following products in your area?” Following this, respondents were asked, “On a scale from
0 to 100, where 0 indicates absolute uncertainty and 100 indicates absolute certainty, how
confident are you in your previous price estimate for the following products?” Respondents
indicated their answers using a slider scale that ranged between 0 and 100 in increments of one.
We asked the uncertainty question in this way because we hypothesized that this question is
easier for subjects to answer than asking them to state percentiles or standard deviations.
Participants were recruited by Survey Sampling International and completed an online
survey hosted on Qualtrics. Additional details on the survey questions, along with descriptive
statistics are provided in the appendix.
To incorporate the subjects’ stated level of reference-price uncertainty into our estimated
model, we specify the standard deviation in equation (6), as follows: 𝜎𝑖𝑗 = √𝛿𝑗 exp(𝛾𝑗
(100−𝑡𝑖𝑗 )
100
),
where 𝛿𝑗 is a parameter to be estimated corresponding to the standard deviation of the referenceprice distribution for good j, 𝑡𝑖𝑗 is individual i’s stated level of certainty in the reference price for
good j which is reverse coded (to convert certainty to uncertainty) and rescaled to range between
0 and 1, and where 𝛾𝑗 is a parameter to be estimated that translates individual-specific
uncertainty, as measured on the survey, to changes in the standard deviation of the subjective
reference-price distribution.
For sake of clarity, Model 4 estimated is estimated as:
18
̅𝑖𝑗
𝑝𝑗−𝑟
𝜑
𝑝𝑗 −𝑟̅𝑖𝑗
(9) 𝐸[𝑉𝑗 ] = 𝛼𝑗 𝛷 (
) 𝑝𝑗 − 𝑟̅𝑖𝑗 + √𝛿𝑗 exp (𝛾𝑗
100−𝑡𝑖𝑗
100−𝑡𝑖𝑗
√𝛿𝑗 exp(𝛾𝑗
)
100
100
(
)
[
(
𝛽𝑗 (1 − 𝛷 (
√𝛿𝑗 exp(𝛾𝑗
)) 𝑝𝑗 − 𝑟̅𝑖𝑗 − √𝛿𝑗 exp (𝛾𝑗
100−𝑡𝑖𝑗
)
100
100
(
)
[
√𝛿𝑗 exp(𝛾𝑗
√𝛿 exp(𝛾
𝑗
𝑗
+
100−𝑡𝑖𝑗
)
100
)]}
100−𝑡𝑖𝑗
)
100 )
+ 𝜏𝑗 .
̅𝑖𝑗
𝑝𝑗 −𝑟
1−𝛷
{
)
̅𝑖𝑗
𝑝𝑗 −𝑟
𝜑
100−𝑡𝑖𝑗
100−𝑡𝑖𝑗
)
100
̅𝑖𝑗
𝑝𝑗 −𝑟
𝛷
{
𝑝𝑗 −𝑟̅𝑖𝑗
√𝛿𝑗 exp(𝛾𝑗
(
√𝛿 exp(𝛾
𝑗
𝑗
100−𝑡𝑖𝑗
)
100 )
]}
The data consist of the experimentally-designed prices used in the choice experiment, 𝑝𝑗 , and the
stated means and uncertainty in the reference price, 𝑟̅𝑖𝑗 and 𝑡𝑖𝑗 . The estimated parameters are 𝛼𝑗 ,
𝛽𝑗 , 𝜏𝑗 , 𝛿𝑗 and 𝛾𝑗 .
Estimation Results and Model Comparisons
Table 5 reports the estimation results for Models 1-4. A likelihood ratio test favors Model 2 over
Model 1 (chi-square value of 135 with 7 degrees of freedom; p-value<0.001). Thus, the data
suggest alternative-specific price effects. For example, respondents are more sensitive to a
change in price for ground beef (-0.718) than they are for steak (-0.405). Qualitatively, Model 3
appears to be an improvement over Model 2 as the results are consistent with loss aversion (i.e.,
𝛼𝑗 < 𝛽𝑗 ) for every good, and the magnitude of the difference is pronounced. In fact, for a couple
of goods (e.g., chicken breast and pasta), 𝛽𝑗 is not significantly different from zero, meaning
people basically ignore price changes when they are below their reference point. Despite this,
the AIC and BIC are higher for Model 3 than for Models 1 and 2, meaning that the reference19
dependent model does not appear to fit the data as well as the more conventional specifications
according to these criteria.
The new specification, Model 4, has the best model fit according to percent of choices
correctly predicted; however, Model 4 has higher AIC and BIC values than Model 2. For Model
4, all the 𝛾𝑗 parameters are positive, as expected, meaning that higher stated reference price
uncertainty on the survey translated into a higher standard deviation around the reference price
distribution. The estimated 𝛿𝑗 parameters are all positive and most are significant, meaning that
there is indeed significant uncertainty in reference prices, which confirms the intuition and need
for this type of model.
One curious aspect associated with model 4 is that several of the 𝛽𝑗 parameters are
positive and statistically significant (although, they are qualitatively small). Taken literally, this
would mean that people prefer paying higher prices when prices are below their reference point.
Such a finding could be rationalized if people associate low prices (prices below reference
points) with signals of low quality. In study 1, quality was likely inferred from brand, and in
study 2, fat-content signaled a key quality attribute; however, in study 3, all products were
unbranded with regard to any quality information, raising the possibility that price was used to
infer quality (Bagwell and Riordan, 1991; Milgrom and Roberts, 1986; Zeithamal, 1988). This
result could also perhaps reflect disadvantageous inequality aversion as described by Fehr and
Schmidt (1999) (i.e., the buyer does not want to take advantage of a seller or to appear “cheap”).
Discussion
Table 6 reports own-price elasticities for the four model specifications. For price increases, we
observe a similar pattern of results as in Studies 1 and 2. Model 4 yields own-price elasticities
20
that are more elastic than those implied by Model 3 but that are more comparable to Models 1
and 2. However, for price decreases, the implied own-price elasticities were most inelastic for
Model 4. The final rows in table 6 report the effect of reducing reference price uncertainty for
each item (assuming all other item’s reference price uncertainty remains unchanged). For each
good, reducing the uncertainty around the product’s reference price increases the likelihood of
purchase. In general, a 1% reduction in the standard deviation of the reference price for a good
translates into a 0.04% to 0.07% increase in choice probability for that good.
Figure 3 illustrates the demand curve for ground beef (one of the most commonly chosen
items in the choice experiment) for the four competing models. Interestingly, models 3 and 4
never intersect, and the market share predictions of model 4 are always less than that for model
3. The bottom of table 5 shows that model 4 is superior to model 3 in terms of every fit critieria
considered. The predictions of model 4 closely mirror that for model 1 for higher price levels,
but for prices less than about $3, model 4 implies a near-linear price response whereas models 1
and 2 imply more elastic price responses.
Conclusions
While models incorporating reference dependent preferences are widely used, it is typically
assumed that the decision maker knows their reference points with certainty. In the case of
price-based reference points, this is a tenuous assumption given that there is ample evidence that
consumers are unaware of past prices paid for products, even for recently purchased products.
Consumers tend to poorly remember previously paid prices (which are often thought to form the
basis for reference points). This paper introduced a new conceptual model that incorporates
reference-price uncertainty into a random utility model of consumer choice and applied the
model to three different studies.
21
The first two studies showed that that the new model uniformly outperformed models that
either ignore reference prices or assumed reference prices were certain. In the third study, the
new model uniformly outperformed the model assuming reference-price certainty and
outperformed the models ignoring reference prices according to one out of three fit criteria.
Overall, the results from the three studies support the idea that reference-price uncertainty is an
important feature of consumer choice. They also suggest that the degree of reference price
uncertainty is linked to the nature of the products under evaluation (e.g., how frequently the
product is being purchased by consumers).
Results from the new model yield predictions that are often starkly at odds with
reference-price models that assume certainty in the reference price. In general, we tend to find
that incorporating reference price uncertainty results in more elastic own-price demand
elasticities than those implied by reference-price models which assume reference-price certainty.
In many cases, allowing for uncertainty more than doubles the size of the own-price elasticity.
We also tend to find, somewhat ironically, that elasticities implied from the new model are more
similar to models which ignore reference prices all together than they are to models that allow
reference prices but assume they are certain. This finding likely results because the new model,
while allowing for asymmetric price responses above and below a reference point, essentially
averages across this kink in the demand curve depending on the degree of reference price
uncertainty. This leads to an “effective” demand curve that is more continuous than the sharply
discontinuous demand curve implied by models that assume the reference price is known with
certainty.
This research highlights a number of issues that warrant future investigation. In
particular, contrasting study 3 with studies 1 and 2 suggests that the way in which reference
22
prices and reference-price uncertainty are measured in a survey may play a role in whether
reference-dependent models (with or without uncertainty) best describe choice; and at this point,
it is unclear which types of questions are likely to be most appropriate. Moving beyond surveybased approaches to the analysis of scanner data, for example, presents challenges in how to
measure reference-price uncertainty. There is a large literature suggesting alternative models for
estimating reference prices from household scanner data. The typical approach relies on pastprices previously paid by the consumer (Briesch et al., 1997; Mazumdar, Raj and Sinha, 2005),
but it is unclear whether reference-price uncertainty should be inferred from the distribution of
previously paid prices, the variance of the error term in the reference-price formation
econometric model, or some other metric like the standard deviation of prices currently being
paid by other consumers in the same geographic area.
Despite these challenges, developments in the literature on endogenous and expectationbased reference point formation appear promising. This paper brought the idea of stochastic
reference points into a model of consumer choice, which also appears to have promise in
explaining phenomena of interest to a broad array of consumer and decision researchers.
23
References
Abeler, J., A. Falk, L. Goette, and D. Huffman. (2011). “Reference Points and Effort Provision.”
American Economic Review 101:470-492.
Adamowicz, W. L., and J.D. Swait. (2012). “Are Food Choices Really Habitual? Integrating
Habits, Variety-Seeking, and Compensatory Choice in a Utility-Maximizing
Framework.” American Journal of Agricultural Economics. 95:17-41.
Bagwell, K., and M.H. Riordan. (1991). “High and Declining Prices Signal Product Quality.”
American Economic Review 81:224-239.
Baucells, M., M. Weber, and F. Welfens. (2011). “Reference-Point Formation and Updating.”
Management Science 57(3):506-519.
Briesch, R.A., L. Krishnamurthi, R. Mazumdar, and S.P. Raj. (1997). “A Comparative Analysis
of Reference Price Models.” Journal of Consumer Research 24:202-214.
Bruno, H.A., H. Che, and S. Dutta. (2012). “Role of Reference Price on Price and Quantity:
Insights from Business-to-Business Markets.” Journal of Marketing Research 49(5):640654.
De los Santos, B., A. Hortacsu and M. R. Wildenbeest. (2012). “Testing Models of Consumer
Search Using Data on Web Browsing and Purchasing Behavior.” American Economic
Review 102:2955-2980.
Erdem, T., G. Mayhew, and B. Sun. (2001). “Understanding the Reference Price Sensitive
Shopper: A Within and Cross-Category Analysis.” Journal of Marketing Research
38:445-457.
Ericson, K.M. M., and A. Fuster, A. (2011). Expectations as Endowments: Evidence on
Reference-Dependent Preferences from Exchange and Valuation Experiments. Quarterly
Journal of Economics 126(4):1879-1907.
Estelami, H. and D.R. Lehmann. (2001). “The Impact of Research Design on Consumer Price
Recall Accuracy: An Integrative Review.” Journal of the Academy of Marketing Science
29:36-49.
Fehr, E., and K.M. Schmidt. (1999). “A Theory of Fairness, Competition, and Cooperation.”
Quarterly journal of Economics 114(3):817-868.
Ghoshal, T., E. Yorkston, J.C. Nunes, and P. Boatwright. (2014). “Multiple Reference Points in
Sequential Hedonic Evaluation: An Empirical Analysis.” Journal of Marketing Research
51:563-577.
24
Han, S., S. Gupta, S., and D.R. Lehmann. (2002). “Consumer Price Sensitivity and Price
Thresholds.” Journal of Retailing 77(4):435-456.
Hasund, K. P., Kataria, M., & Lagerkvist, C. J. (2011). Valuing public goods of the agricultural
landscape: a choice experiment using reference points to capture observable
heterogeneity. Journal of Environmental Planning and Management 54(1), 31-53.
Hu, W., W.L. Adamowicz, and M.M. Veeman. (2006). “Labeling Context and Reference Point
Effects in Models Of Food Attribute Demand.” American Journal of Agricultural
Economics 88(4):1034-1049.
Hardie, B.G., E.J. Johnson, and P.D. Fader. (1993). “Modeling Loss Aversion and Reference
Dependence Effects on Brand Choice.” Marketing Science 12(4):378-394.
Jensen, B.B. and K.G. Grunert. (2014). “Price Knowledge during Grocery Shopping: What We
Learn and What We Forget.” Journal of Retailing. forthcoming.
Kahneman, D., and A. Tversky. (1979). “Prospect Theory: An Analysis of Decision Under
Risk.” Econometrica 47:263-291.
Kalyanaram, G., and R.S. Winer. (1995). “Empirical Generalizations from Reference Price
Research.” Marketing Science 14:G161-G169.
Kim, J. B., P. Albuquerque, and B. J. Bronnenberg. 2010. “Online Demand under Limited
Consumer Search.” Marketing Science 29:1001-1023.
Kőszegi, B., and M. Rabin. (2006). “A Model of Reference-Dependent Preferences.” Quarterly
Journal of Economics 121:1133-1165.
Louviere, J.J., D.A. Hensher, and J.D. Swait. (2000). “Stated Choice Methods: Analysis and
Applications.” Cambridge University Press.
Mabit, S.L. and M. Fosgerau. (2011). “Demand for Alternative-Fuel Vehicles When Registration
Taxes Are High.” Transportation Research Part D: Transport and Environment
16(3):225-231.
Mazumdar, T., S.P. Raj, and I. Sinha. (2005). “Reference Price Research: Review and
Propositions. Journal of Marketing 69(4):84-102.
Mehta, N., D. Rajiv, and K. Srinivasan. (2003). “Price Uncertainty and Consumer Search: A
Structural Model of Consideration Set Formation.” Marketing Science 22(1):58-84.
Milgrom, P., and J. Roberts, J. (1986). “Price and Advertising Signals of Product Quality.”
Journal of Political Economy 94:796-821.
25
Moon, S., G. J. Russell, and S.D. Duvvuri. (2006). “Profiling the Reference Price Consumer.”
Journal of Retailing 82(1):1–11
Neumann, N., and U. Böckenholt. (2014). “A Meta-analysis of Loss Aversion in Product
Choice.” Journal of Retailing 90:182-197.
Nicolau, J.L. (2011). “Differentiated Price Loss Aversion in Destination Choice: The Effect Of
Tourists’ Cultural Interest.” Tourism Management 32(5):1186-1195.
Pulter, D.S. (1992). “Incorporating Reference Price Effects into a Theory of Consumer Choice.”
Marketing Science 11:287-309.
Rose, J.M. and L. Masiero. (2010). “A Comparison of the Impacts of Aspects of Prospect Theory
in WTP/WTA Estimated in Preference and WTP/WTA Space.” European Journal of
Transport and Infrastructure Research 10(4):330-346.
Scarpa, R., S. Ferrini, and K. Willis. (2005). “Performance of Error Component Models for
Status-Quo Effects in Choice Experiments.” in Applications of Simulation Methods in
Environmental and Resource Economics (R. Scarpa and A. Alberini, eds.). Springer
Netherlands 6:247-273.
Terui, N. and W.D. Dahana. (2006). “Research Note-Estimating Heterogeneous Price
Thresholds.” Marketing Science 25(4):384-391.
Tversky, A., and D. Kahneman. (1991). “Loss Aversion in Riskless Choice: A ReferenceDependent Model.” Quarterly Journal of Economics 106(4):1039-1061.
Urbany, J.E. and P.R. Dickson. (1991). “Consumer Normal Price Estimation: Market versus
Personal Standards.” Journal of Consumer Research 18(1991):45-51.
Weaver, R, and S. Frederick. (2012). “A Reference Price Theory of the Endowment Effect.”
Journal of Marketing Research 49:696-707.
Winer, R. S. (1986). “A Reference Price Model of Brand Choice for Frequently Purchased
Products.” Journal of Consumer Research 13:250-256.
Vanhuele M. and X. Drèze. (2002). “Measuring the Price Knowledge Shoppers Bring to the Store.”
Journal of Marketing 66:72−85.
Zeithaml, V.A. (1988). “Consumer Perceptions of Price, Quality, and Value: A Means-End
Model and Synthesis of Evidence.” Journal of Marketing 52:2-22.
Zhao, J. and C.L. Kling. (2004). “Willingness to Pay, Compensating Variation, and the Cost of
Commitment.” Economic Inquiry 42(3):503-517.
26
$4.50
$4.00
$3.50
Model 1
$3.00
Model 2
Model 3
$2.50
Model 4
$2.00
$1.50
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Market share
Figure 1. Implied Demand Curves for Heinz Ketchup from Four Competing Models
27
0.9
1
$4.50
$4.00
$3.50
Model 1
Model 2
$3.00
Model 3
Model 4
$2.50
$2.00
$1.50
0
0.1
0.2
0.3
0.4
0.5
0.6
Market share
Figure 2. Implied Demand Curves for Whole Milk from Four Competing Models
28
0.7
0.8
$5.00
$4.50
$4.00
Model 1
Model 2
$3.50
Model 3
Model 4
$3.00
$2.50
$2.00
0
0.05
0.1
0.15
0.2
0.25
0.3
Market share
Figure 3. Implied Demand Curves for Ground Beef from Four Competing Models
29
0.35
Table 1. Estimates and Model Comparisons for Ketchup
Parameter
Model 1
Brand 1 (Red Gold)
4.690*a (0.333)b
𝜏1
-1.184* (0.025)
𝛼1
-1.184* (0.025)
𝛽1
Brand 2 (Del Monte)
5.499* (0.332)
𝜏2
-1.184* (0.025)
𝛼2
-1.184* (0.025)
𝛽2
Brand 3 (Heinz)
7.746* (0.335)
𝜏3
-1.184* (0.025)
𝛼3
-1.184* (0.025)
𝛽7
Brand 4 (Hunt’s)
6.305* (0.332)
𝜏4
-1.184* (0.025)
𝛼4
-1.184* (0.025)
𝛽4
Error Component
σ e2
9.732* (1.668)
LLF
# parms
% correctc
AICc
BICc
-4776
6
61.06%
9564
9584
Model 2
Model 3
Model 4
4.677* (0.371)
-1.183* (0.086)
-1.183* (0.086)
1.832* (0.315)
-1.492* (0.166)
-0.759* (0.104)
1.898* (0.323)
-1.491* (0.173)
-0.726* (0.114)
6.178* (0.357)
-1.512* (0.071)
-1.512* (0.071)
2.803* (0.307)
-2.030* (0.147)
-0.618* (0.082)
3.012* (0.314)
-2.184* (0.161)
-0.482* (0.092)
7.372* (0.342)
-1.067* (0.034)
-1.067* (0.034)
4.758* (0.302)
-1.507* (0.054)
-0.615* (0.070)
4.875* (0.306)
-1.575* (0.060)
-0.503* (0.077)
6.361* (0.344)
-1.215* (0.047)
-1.215* (0.047)
3.474* (0.303)
-1.504* (0.082)
-0.615* (0.070)
3.632* (0.307)
-1.610* (0.090)
-0.503* (0.077)
9.605* (1.650)
7.441* (1.332)
7.544* (1.346)
-4757
9
61.06%
9532
9563
-4660
13
62.57%
9347
9391
-4652
13
62.72%
9330
9373
a
One star reflects statistical significance at the 0.05 level or lower.
Numbers in parentheses are standard errors.
c
Bolded values indicate best model according to the respective criteria.
b
30
Table 2. Own-Price Arc Elasticities Implied by Competing Models for Ketchup
Model 1
Model 2
Price Increase from $3 to $4
Brand 1
-2.061
-2.059
Brand 2
-2.033
-2.309
Brand 3
-1.245
-1.090
Brand 4
-1.966
-1.995
a
Price Decrease from $3 to $2
Brand 1
-6.138
-6.107
Brand 2
-5.430
-8.393
Brand 3
-0.893
-0.785
Brand 4
-4.191
-4.319
Decrease in reference price uncertainty b
Brand 1
0
0
Brand 2
0
0
Brand 3
0
0
Brand 4
0
0
Model 3
Model 4
-1.614
-1.337
-1.147
-1.275
-2.208
-2.468
-1.548
-2.163
-3.513
-2.453
-0.930
-1.968
-5.469
-5.951
-0.939
-3.363
0
0
0
0
-0.001
0.020
-0.025
0.012
a
Although a price decrease was simulated, we the values are reported as negatives to reflect the more common way
own-price elasticities of demand are reported.
b
Simulated by determining the percent change in probability of purchase for each brand induced by a 25% reduction
in the standard deviation of reference price, expressed in elasticity terms (i.e., the % change in probability of
purchase for a 1% reduction in reference price uncertainty).
31
Table 3. Estimates and Model Comparisons for Milk
Parameter
Model 1
Option 1 (Skim)
3.816*a (0.186)b
𝜏1
-1.055* (0.022)
𝛼1
-1.055* (0.022)
𝛽1
Option 2 (2% reduced fat)
5.188* (0.186)
𝜏2
-1.055* (0.022)
𝛼2
-1.055* (0.022)
𝛽2
Option 3 (whole)
4.899* (0.185)
𝜏3
-1.055* (0.022)
𝛼3
-1.055* (0.022)
𝛽7
Option 4 (1 % reduced fat)
4.276* (0.184)
𝜏4
-1.055* (0.022)
𝛼4
-1.055* (0.022)
𝛽4
Error Component
σ e2
4.231* (0.655)
LLF
# parms
% correctc
AICc
BICc
-6478
6
48.52%
12968
12889
Model 2
Model 3
Model 4
3.684* (0.230)
-1.008* (0.054)
-1.008* (0.054)
1.109* (0.18)
-1.524* (0.104)
-0.391* (0.082)
1.148* (0.186)
-1.490* (0.108)
-0.394* (0.089)
5.548* (0.208)
-1.170* (0.037)
-1.170* (0.037)
2.288* (0.172)
-1.669* (0.069)
-0.776* (0.062)
2.413* (0.176)
-1.762* (0.076)
-0.684* (0.068)
4.348* (0.206)
-0.877* (0.036)
-0.877* (0.036)
2.025* (0.173)
-1.497* (0.071)
-0.407* (0.071)
2.106* (0.176)
-1.538* (0.076)
-0.282* (0.079)
4.642* (0.222)
-1.183* (0.050)
-1.183* (0.050)
1.516* (0.175)
-1.498* (0.088)
-0.407* (0.071)
1.685* (0.180)
-1.638* (0.097)
-0.282* (0.079)
4.236* (0.656)
3.859* (0.594)
3.870* (0.596)
-6457
9
48.86%
12931
12962
-6312
13
51.13%
12649
12693
-6309
13
51.22%
12645
12689
a
One star reflects statistical significance at the 0.05 level or lower.
Numbers in parentheses are standard errors.
c
Bolded values indicate best model according to the respective criteria.
b
32
Table 4. Own-Price Arc Elasticities Implied by Competing Models for Milk
Model 1
Model 2
Price Increase from $3.50 to $4.50
Option 1
-2.203
-2.139
Option 2
-1.896
-2.063
Option 3
-2.015
-1.739
Option 4
-2.152
-2.316
Price Decrease from $3.50 to $2.50 a
Option 1
-5.073
-4.722
Option 2
-2.512
-2.887
Option 3
-3.179
-2.469
Option 4
-4.400
-5.262
Decrease in reference price uncertainty b
Option 1
0
0
Option 2
0
0
Option 3
0
0
Option 4
0
0
Model 3
Model 4
-1.120
-1.564
-1.377
-1.108
-2.467
-2.454
-2.369
-2.508
-1.753
-2.081
-1.893
-1.621
-4.059
-2.908
-2.960
-3.506
0
0
0
0
0.001
0.003
0.000
0.005
a
Although a price decrease was simulated, we the values are reported as negatives to reflect the more common way
own-price elasticities of demand are reported.
b
Simulated by determining the percent change in probability of purchase for each brand induced by a 25% reduction
in the standard deviation of reference price, expressed in elasticity terms (i.e., the % change in probability of
purchase for a 1% reduction in reference price uncertainty).
33
Table 5. Estimates and Model Comparisons for Meal Choices
Parameter
Model 1
Model 2
Model 3
Model 4
4.847*a (0.197)b
-0.584* (0.011)
-0.584* (0.011)
-----
5.230* (0.212)
-0.718* (0.03)
-0.718* (0.03)
-----
3.364* (0.193)
-0.980* (0.054)
-0.028* (0.012)
-----
4.363*
-1.201*
0.086*
1.987*
2.831*
6.063* (0.206)
-0.584* (0.011)
-0.584* (0.011)
-----
5.012* (0.277)
-0.405* (0.033)
-0.405* (0.033)
-----
2.612* (0.196)
-0.380* (0.035)
-0.100* (0.011)
-----
3.608* (0.276)
-0.538* (0.044)
-0.004 (0.020)
6.643 (3.552)
3.112* (0.600)
4.446* (0.198)
-0.584* (0.011)
-0.584* (0.011)
-----
4.542* (0.225)
-0.613* (0.036)
-0.613* (0.036)
-----
2.743* (0.196)
-0.752* (0.060)
-0.049* (0.014)
-----
3.851*
-0.964*
0.066*
1.707*
3.811*
3.682* (0.197)
-0.584* (0.011)
-0.584* (0.011)
-----
3.616* (0.209)
-0.552* (0.037)
-0.552* (0.037)
-----
2.357* (0.196)
-0.759* (0.087)
-0.076* (0.011)
-----
3.375* (0.239)
-0.792* (0.078)
0.026 (0.017)
3.138* (1.393)
3.574* (0.437)
Ground Beef
𝜏1
𝛼1
𝛽1
𝛿1
𝛾1
(0.231)
(0.058)
(0.018)
(0.666)
(0.338)
Steak
𝜏2
𝛼2
𝛽2
𝛿2
𝛾2
Pork Chop
𝜏3
𝛼3
𝛽3
𝛿3
𝛾3
(0.242)
(0.064)
(0.021)
(0.712)
(0.432)
Ham
𝜏4
𝛼4
𝛽4
𝛿4
𝛾4
Table 5 continued on next page
34
Table 5 continued
Parameter
Model 1
Chicken Breast
5.333* (0.196)
𝜏5
-0.584* (0.011)
𝛼5
-0.584* (0.011)
𝛽5
--𝛿5
--𝛾5
Chicken Wing
3.615* (0.195)
𝜏6
-0.584* (0.011)
𝛼6
-0.584* (0.011)
𝛽6
--𝛿6
--𝛾6
Beans and Rice
3.481* (0.196)
𝜏7
-0.584* (0.011)
𝛼7
-0.584* (0.011)
𝛽7
--𝛿7
--𝛾7
Pasta
4.281* (0.200)
𝜏8
-0.584* (0.011)
𝛼8
-0.584* (0.011)
𝛽8
--𝛿8
--𝛾8
Error Component
σ e2
8.068* (0.792)
LLF
# parms
% correctc
AICc
BICc
-16876
10
16.13%
33773
33822
Model 2
Model 3
Model 4
5.652* (0.204)
-0.696* (0.023)
-0.696* (0.023)
-----
4.013* (0.192)
-0.986* (0.041)
-0.003 (0.011)
-----
4.913*
-1.186*
0.107*
1.584*
3.025*
(0.217)
(0.045)
(0.017)
(0.435)
(0.279)
3.716* (0.204)
-0.655* (0.041)
-0.655* (0.041)
-----
2.758* (0.194)
-1.124* (0.094)
-0.072* (0.011)
-----
3.917*
-1.186*
0.041*
1.736*
3.643*
(0.235)
(0.084)
(0.017)
(0.654)
(0.357)
3.248* (0.201)
-0.417* (0.032)
-0.417* (0.032)
-----
2.775* (0.195)
-0.510* (0.037)
-0.040* (0.012)
-----
3.773*
-0.820*
0.050*
2.310*
3.407*
(0.251)
(0.060)
(0.018)
(1.153)
(0.497)
3.602* (0.236)
-0.382* (0.037)
-0.382* (0.037)
-----
2.855* (0.200)
-0.510* (0.037)
-0.008 (0.020)
-----
3.627* (0.247)
-0.664* (0.048)
0.081* (0.025)
1.132 (0.757)
4.262* (0.720)
8.110* (0.796)
7.788* (0.776)
7.736* (0.777)
-16809
17
16.13%
33652
33735
-17187
25
22.63%
34423
34547
-16792
41
23.19%
33666
33868
a
One star reflects statistical significance at the 0.05 level or lower.
Numbers in parentheses are standard errors.
c
Bolded values indicate best model according to the respective criteria.
b
35
Table 6. Own-Price Arc Elasticities Implied by Competing Models for Meal Choices
Model 1
Model 2
Price increase from mid to highest level in design
Ground beef
-1.269
-1.459
Steak
-2.431
-1.865
Pork chop
-1.403
-1.450
Ham
-0.996
-0.959
Chicken breast
-1.088
-1.244
Chicken wings
-0.644
-0.696
Beans and rice
-0.747
-0.585
Pasta
-1.513
-1.114
Price decrease from mid to lowest level in design a
Ground beef
-2.304
-3.088
Steak
-4.939
-3.029
Pork chop
-3.550
-3.570
Ham
-2.066
-1.908
Chicken breast
-1.576
-1.969
Chicken wings
-1.230
-1.441
Beans and rice
-1.506
-0.958
Pasta
-3.215
-1.813
b
Decrease in reference price uncertainty
Ground beef
0
0
Steak
0
0
Pork chop
0
0
Ham
0
0
Chicken breast
0
0
Chicken wings
0
0
Beans and rice
0
0
Pasta
0
0
a
Model 3
Model 4
-0.640
-1.205
-0.973
-0.332
-0.611
-0.169
-0.226
-0.917
-1.201
-1.241
-1.141
-0.543
-1.010
-0.545
-0.574
-1.213
-1.320
-1.822
-3.068
-0.654
-1.075
-0.359
-0.444
-1.781
-1.247
-1.353
-1.581
-0.533
-0.869
-0.559
-0.626
-1.593
0
0
0
0
0
0
0
0
0.062
0.056
0.072
0.062
0.046
0.075
0.066
0.046
Although a price decrease was simulated, we the values are reported as negatives to reflect the more common way
own-price elasticities of demand are reported.
b
Simulated by determining the percent change in probability of purchase for each brand induced by a 25% reduction
in the standard deviation of reference price for that brand, expressed in elasticity terms (i.e., the percent change in
probability of purchase for a 1% reduction in reference price uncertainty)
36