Apple Inc. v. Samsung Electronics Co. Ltd. et al

Filing 1061

Administrative Motion to File Under Seal Samsung's Reply In Support of Motion To Exclude Opinions of Certain of Apple's Experts filed by Samsung Electronics America, Inc.(a New York corporation), Samsung Electronics Co. Ltd., Samsung Telecommunications America, LLC(a Delaware limited liability company). (Attachments: #1 Samsung's Reply In Support of Motion To Exclude Opinions of Certain of Apple's Experts, #2 Declaration Declaration of Joby Martin, #3 Exhibit 2, #4 Exhibit 3, #5 Proposed Order Granting Samsung's Administrative Motion to File Documents Under Seal)(Maroulis, Victoria) (Filed on 6/7/2012)

EXHIBIT 3

How Much Does the Market Value an
Improvement in a Product Attribute?
Elie Ofek  V. Srinivasan
Harvard Business School, Soldiers Field, Boston, Massachusetts 02163
Graduate School of Business, Stanford University, Stanford, California 94305-5015
[e-mail]  [e-mail]

A

firm contemplating improvements to its product attributes would be interested in
the dollar value the market attaches to any potential product modification. In this
paper, we derive a measure of market value such that the comparison of the measure
against the incremental unit cost of the attribute improvement is key in deciding whether or
not the attribute improvement is profitable. Competition from other brands, the potential
for market expansion, and heterogeneity in customer preference structures are explicitly
modeled using the multinomial logit framework. The analysis yields a closed form
expression for the market’s value for an attribute improvement (MVAI). A key result we
obtain is that customers should be differentially weighted based on their probability of
purchasing the firm’s product. In particular, customers who exhibit a very high or very
low probability of choosing the firm’s product should receive less weight in detemining
MVAI. Because the probability of choice varies across products, the answer to the question
of how much the market values an improvement depends on which firm is asking the
question. It is shown that customers whose utilities have a greater random component
should be weighted less. Furthermore, the measure developed is robust to the influence of
outliers in the sample. An empirical illustration of the MVAI measure in the context of
a new product development study is provided. The study illustrates the advantages of the
proposed measure over currently used approaches and explores the possibility of competitive price reactions.
(New Product Development; Product Positioning; Multibrand Competition; Conjoint Analysis)

1. Introduction
In many product markets, firms often desire to
modify their product attributes. Evolving consumer
preferences, advances in technological capabilities,
changes in manufacturing costs, and competition
from other brands drive firms to consider improving
product characteristics. While some companies attempt to develop radically innovative products, new
product activity often involves the modification of
an existing product. Typically, value adding modifi-

MARKETING SCIENCE Ó 2002 INFORMS
Vol. 21, No. 4, Fall 2002, pp. 398–411

cations entail offering more of a desirable attribute
or less of an undesirable one.1 Such product changes
have both cost and demand implications, and require reevaluating pricing decisions as well. While,
1

In reality, there may also be cases where firms wish to accomplish the opposite, i.e., offer less of a desirable attribute or more
of an undesirable one (perhaps because of a cost increase). The
analysis we present is general enough to handle these cases as
well. We have framed the problem in terms of product improvement as it is the most common form of product modification
(Griffin 1997).

0732-2399/02/2104/0398/$05.00
1526-548X electronic ISSN

HOW MUCH DOES THE MARKET VALUE AN IMPROVEMENT IN A PRODUCT ATTRIBUTE?

in general, firms know their own cost structures, assessing demand sensitivity to product changes can
be difficult, especially when the market is comprised
of customers heterogeneous in preferences and there
are several competing brands. Nonetheless, accurate
assessment of the market’s response to any such attribute improvement is essential to a firm in the
product planning phase, and for effective pricing
and forecasting.
To explore this issue in greater detail, we seek to
establish a market-level analog to an individual customer’s value for an improvement in a product attribute. The latter quantity is usually defined as the
increase in price needed to offset an incremental
change in an attribute level, so that the customer’s
overall utility for a particular product remains constant. This calculation is straightforward once the
parameters of the utility function for that individual
have been estimated by conjoint analysis (Green and
Srinivasan 1990). Yet when dealing with a heterogeneous set of customers, all potentially relevant for
the firm’s market share, it is not obvious how to aggregate these individual parameters to form a single
market-level valuation index. From a managerial
perspective, it is the aggregate measure that is relevant for product planning rather than individuallevel measures.
In this paper, we theoretically derive such a market-level valuation measure by looking at the profit
change a firm can expect from an incremental improvement in a product attribute when setting its
price optimally. We define our measure as the market’s value for an attribute improvement (MVAI)
and show it has three major advantages over current approaches. First, MVAI has the managerially
attractive property that it can be compared to the
incremental unit cost of the attribute improvement
to determine the profitability of the product modification. Second, it provides a conceptually sound
method for calculating the market’s value for
a product attribute compared to the commonly
used method of averaging customer-level willingness-to-pay measures. Third, by deriving an analytic expression in the context of the standard
multinomial logit framework, we are able to offer

MARKETING SCIENCE/Vol. 21, No. 4, Fall 2002

many valuable insights into the factors that affect
the market’s value for an attribute improvement.
Specifically, we find that the market’s value for an
improvement is not obtained as an average of individual-level measures, i.e., by averaging the dollar
amount needed to keep each customer’s preference
constant. Rather, the precise formula involves the
ratio of two separate and weighted sums across customers: one related to the importance of the attribute and the other to the importance of price. It is
shown that customers should be differentially
weighted based on their probability of purchasing
the focal product. A customer with an extreme
probability (either approaching zero or one) of purchasing the focal product is relatively insensitive to
that product’s attribute and/or price modifications
and, hence, should be given less weight. Because
the choice probabilities vary across products, the
market’s value for an improvement in a product attribute depends on which competitive product is
asking the question. The approach discounts customers whose utilities have a greater error component and are therefore less susceptible to attribute
and/or price changes. In addition, the measure is
fairly robust to the influence of outliers in the data,
making it unnecessary to drop observations
or worry about extreme results. In the context of
a new product development study, this paper provides an empirical illustration of the advantages of
the proposed measure over currently used approaches and explores competitive price reactions.
The rest of the paper is organized as follows: We
first derive a general formulation of the market’s
value for an incremental improvement in a product
attribute for a firm assessing the profitability of
a product modification. Next, using the standard
logit framework to link individual preference
parameters to market shares, we calculate an explicit expression for MVAI and highlight the implications for how individual parameters should be
weighted and aggregated. An empirical illustration
of the proposed measure is then provided that also
incorporates competitive price reactions. We conclude with managerial implications of the MVAI
measure.

399

OFEK AND SRINIVASAN
How Much Does the Market Value an Improvement in a Product Attribute?

2. Firm Profitability and
the Market’s Value for
an Improvement in
a Product Attribute
An important financial criterion for measuring the
success of a product development effort is whether
it would result in increased profits. For the type of
product modification considered in this paper, the
relevant question becomes: What determines whether
an incremental improvement in a product attribute
will result in an increase in profits? We turn to examine this question assuming that, subsequent to any
product modification, a firm would adjust its price to
achieve maximal profits. As we shall see, answering
the above question will allow us to derive a measure
of the market’s value for an incremental improvement in a product attribute. We begin by analyzing
the pricing action of the product modifying firm, assuming that no other firm reacts to its actions (this
case is realistic if, in the short run, other firms in the
product market are committed to their prices, and it
also applies to the monopoly case). Such an analysis
is a starting point for determining the profitability of
an attribute change. We consider competitive price
reactions in §4.3 (and Appendix) and show that our
proposed measure is relevant in such contexts as
well. In this respect, our work is related to the marketing literature on competitive positioning (e.g.,
Hauser and Shugan 1983, Ansari et al. 1994). Such
papers analyze firm pricing and product attribute
decisions in competitive settings and incorporate
customer heterogeneity in the form of probabilistic
distributions. Our paper contributes to the understanding of how such decisions should be made by
analyzing and providing insights on the way individual-level data can be weighted and aggregated
so that a single, theoretically meaningful measure of
market value for an attribute improvement can be
obtained.
2.1. Model
Consider a given product market consisting of J
products each offered by a different firm. Let each
of the J products be defined by a vector of K product

400

attributes ~, with the level of attribute k for alternaxj
tive j denoted xjk. Denote the market share of firm
(product) j as mj and its unit price as pj. We allow
for the possibility of an outside good with market
P
share m0, such that m0 1 Jj¼1 mj 5 1. Denote by Q
the maximum sales potential for the product category,
so that the actual sales of the category of J products
P
are Q(1 2 m0) 5 Q Jj¼1 mj. The outside good permits the market (i.e., sales) for the category of J products to expand or contract depending on the J
products’ attributes and prices. Furthermore, assume
that market shares are differentiable in prices and attribute values. To set up a framework for firm optimizing behavior, let firm j’s profit function pj be
x
pj ¼ Qmj ðpj À cj ð~j ÞÞ;

ð1Þ

where cj(~) is firm j’s variable cost of producing
xj
each unit, which depends on the particular attribute
levels offered by product j (for notational convenience, we suppress this dependence and, from here
on, denote the variable cost as cj).2 Let all market
shares satisfy the following standard properties of
price competition with differentiated goods: @mj/@pj
, 0, @mj9/@pj . 0 "j9 6¼ j. In particular, @m0/@pj .
0, so that aggregate category sales, given by Q(1 2
m0), depend on the firms’ prices (the outside good
also allows for the special case of a monopoly ( J 5
1) whose sales depend on its product attributes and
price.) In §3, we elaborate on how market shares,
which satisfy the above properties, can be determined as a function of customer preferences and
product attribute levels. We further assume that the
profit functions in (1) satisfy @ 2 pj /@p2 , 0, so that
j
we are guaranteed an interior pricing solution.
Optimizing behavior by firm j means that it sets
a price to satisfy
@pj
¼ 0:
@pj

ð2Þ

2

To simplify analysis, we assumed that there are no economies (or
diseconomies) of scale so that the variable cost cj does not depend
on the sales level (Qmj) of product j. For expositional covenience,
we also assumed that there are no fixed costs.

MARKETING SCIENCE/Vol. 21, No. 4, Fall 2002

OFEK AND SRINIVASAN
How Much Does the Market Value an Improvement in a Product Attribute?

Substituting (1) into (2) yields the following firstorder condition:
mj þ

@mj
ðpj À cj Þ ¼ 0:
@pj

ð3Þ

We can now examine the total effect on profitability
triggered by the change in attribute xjk. This is given by
dpj
@pj @pj dpÃ
j
¼
þ
;
dxjk @xjk @pj dxjk

ð4Þ

where pà is the optimal price according to (3). From
j
the first-order condition in (2), we know the second
term on the right-hand side of (4) is zero. Thus, (4)
can be rewritten as
dpj
@pj Q@mj
@cj
¼
¼
ðpj À cj Þ À Qmj
:
dxjk @xjk
@xjk
@xjk

ð5Þ

We see from (5) that when firm j considers only its
actions and sets a price to maximize profits, the condition for the attribute change to be profitable is
À

@mj =@xjk @cj
.
:
@mj =@pj @xj

ð6Þ

The left-hand side of (6) is a ratio of the change
in market share the modifying firm can expect
from improving a product attribute, divided by the
change in market share due to repricing. As this
ratio reflects aggregate demand sensitivity for the
incremental attribute and price changes, we define
it to be the market’s value for attribute improvement
(MVAI):
MVAI ¼ À

@mj =@xjk
:
@mj =@pj

ð7Þ

The inequality in (6) has an appealing interpretation;
it states that the profitability of each unit sold
depends on how much the market value for the
proposed improvement exceeds the marginal cost of

MARKETING SCIENCE/Vol. 21, No. 4, Fall 2002

dmj ¼

@mj
@mj
dxjk þ
dpj :
@xjk
@pj

ð8Þ

Suppose we wish to leave firm j’s market share unaltered; i.e., we require dmj 5 0. It follows from (8)
and (7) that
dpj
@mj =@xjk
¼À
¼ MVAI:
dxjk
@mj =@pj

Substituting from (3) we obtain


dpj
@mj =@xjk @cj
¼ Qmj À
À
:
dxjk
@mj =@pj @xjk

executing it (aside from any one-time fixed costs).
The decision on whether to implement a particular
product change takes on an explicit benefit versus
cost form. Furthermore, this ‘‘benefit’’ to the firm is
fundamentally demand driven.
An alternative interpretation of MVAI in (7) comes
from the following observation. The total differential
of firm j’s market share with respect to attribute k
and price pj is

ð9Þ

Thus, MVAI is also the incremental price the firm
would charge per unit improvement in the product
attribute (assumed to be infinitesimal) if it were to
hold market share (or sales) constant.
Interpreting MVAI as the incremental price change
(divided by the attribute change) that leaves market
share unaltered suggests a computational method
for determining MVAI. First, change the product
attribute in the conjoint simulator and then search
over the price change that would leave market share
unaltered. An efficient way for conducting such
a search on price is the method of interval bisection
(Wagner 1975, p. 539). As an approximation, one
could use this computational method in conjunction
with the max-choice rule, in which case the market
share for each alternative is obtained by counting all
individuals (weighted based on the quantities they
purchase in the product category) whose preference
for the alternative is highest and dividing by the total number of individuals. The drawback of the computational method is the lack of a closed-form
solution for MVAI. In the next section, we provide
such a closed-form expression for MVAI in the context of the logit model. The expression also provides
valuable insights into the determinants of MVAI.

401

OFEK AND SRINIVASAN
How Much Does the Market Value an Improvement in a Product Attribute?

3. A Logit Model-Based Expression
for the Market’s Value for
an Attribute Improvement
Thus far, we provided a general form for the market’s value for a product improvement and established its importance for assessing the profitability
of a proposed attribute change. As MVAI is a function of market share, it is obviously related to the
change in demand for the modified product. It is
not clear, however, what customer-level parameters
are required to determine it, how such parameters
should be aggregated, and what the dependence on
the current set of competing products is. To shed
light on these issues, we proceed as follows: We begin by specifying a simple model of customer multiattribute preferences, which is then used to obtain
market shares within the standard logit model framework. These, in turn, are used to derive a closed
form expression for MVAI. We highlight the analytic
insights gained from this proposed measure and discuss its practical benefits.
3.1. Individual Preferences
Since our primary interest is the trade-off between
attribute levels and price, as well as the resulting
impact on market behavior, it is only natural to operate in the setting of multiattribute preference models
(Green and Srinivasan 1990). As such, an individual’s deterministic preference for any alternative can
be written as a sum of part worths and is given by
tij ¼

Kþ1
X

fki ðxjk Þ;

ð10Þ

k¼1

where
tij 5 the (deterministic) utility individual i attaches
to alternative j;
xjk 5 level of attribute k for alternative j (there are
K 1 1 attributes, including price); and
i
fk 5 the part-worth function relating level of attribute k into utility units for individual i.
For expositional ease, we will assume the functions
i
fk conform to the vector (or linear) model; i.e., we can
replace them with individual attribute weights wik .

402

As is common with most multiattribute preference
models, we treat price as a separable determinant of
utility. Consistent with previous notation, we denote
the price of alternative j by pj and its coefficient by
(2wip ). Equation (10) can now be written
tij ¼

K
X

!
wik xjk

À wip pj ;

ð11Þ

k¼1

where wip . 0 (note that in (11) price would relate
negatively to utility). We assume that the parameters
{wik }, wip and the randomness parameter (see §3.2) are
estimated by conjoint analysis (Green and Srinivasan
1978). In the Technical Appendix,3 we derived some
of the major results using the more general formulation, as in (10).
3.2.

From Customer Preferences
to Market Shares
To obtain market shares from customer preferences,
we first link overall utility for an alternative with
a choice probability. As we shall see, this approach
yields closed form expressions for both market
shares and MVAI.
Linking overall preference for an alternative with
a choice probability can be achieved by invoking
random utility theory (Ben-Akiva and Lerman 1985,
pp. 60–66). Thus, we can regard the multiattribute
preference model discussed earlier as the deterministic (or systematic) component of utility, to which
a random component is added. Thus, the utility individual i assigns alternative j is uij 5 tij 1 eij , where
tij is the deterministic component and eij is the random component. The deterministic component is obtained from (11) (or (10) in the more general case).
The stochastic components eij are assumed to be
independent across j and distributed Gumbel with
cumulative distribution function F(eij ) 5 exp[2exp(2lieij )], where li . 0 is individual i’s scale
parameter, which is inversely related to the variance
of the random component Var(eij ) 5 p2/6l2 .
i
3

The Technical Appendix is available at the INFORMS Marketing
Science website at http://www.informs.org.

MARKETING SCIENCE/Vol. 21, No. 4, Fall 2002

OFEK AND SRINIVASAN
How Much Does the Market Value an Improvement in a Product Attribute?

From this specification of random utility, choice
probabilities can easily be derived using the multinomial logit model (McFadden 1974, Ben-Akiva and
Lerman 1985, pp. 103–107). Hence, the probability of
individual i choosing alternative j, when J alternatives (and an outside good) are available, takes the
familiar form
hij ¼

expðli tij Þ
;
P
expðli ti0 Þ þ Jj9¼1 expðli tij9 Þ

¼

ð13Þ

It is straightforward to establish that market shares
thus defined satisfy all the properties required in §2
(namely, @mj/@pj , 0, @mj9/@pj . 0 "j9 6¼ j, and
@m0/@pj . 0).
3.3.

Calculating Market Value for an Attribute
Improvement
Having established how market shares are formed
from individual preferences, we are now ready to
apply this framework to derive an explicit expression
for MVAI. Using (11)–(13), (7) can be shown to be

MARKETING SCIENCE/Vol. 21, No. 4, Fall 2002

P
¼

qi li hij ð1
Pi
i
i qi li hj ð1

P i i
i aj wk
¼P i i;
i
i
À hj Þwp
i aj wp

À hij Þwik

ð14Þ

where aij represent customer weights for the jth product and are given by
aij ¼ qi li hij ð1 À hij Þ:

1 X i
qi hj
Q i
qi expðli tij Þ
1 X
P
:
Q i expðli ti0 Þ þ j9 expðli tij9 Þ

@mj =@xjk
@mj =@pj

ð12Þ

where ti0 is customer i’s (deterministic) utility for the
substitute outside good. Let qi denote the (exogenously specified) maximal purchase quantity for cusP
tomer i. Consistent with previous notation, Q 5 i qi
is the maximum product category sales. The
expected quantity each individual purchases in the
category of J products is thus qi(1 2 hi0 ). As the
probability of choosing the outside good (or no purchase) is a function of the prices and attributes of all
J products in the market, the total sales of the J
products in the category generally increase when
any firm decreases price or improves its product
attributes.
The calculation of market shares is now straightforward. For each alternative, add the individual
probabilities of choosing it, weighted by the appropriate purchase quantities
mj ¼

MVAI ¼ À

ð15Þ

3.4. Properties of the MVAI Expression
This section highlights the major insights from the
previous analysis, in particular, the properties of
Equations (14)–(15).
3.4.1. The Aggregation of Individual Influences.
Let us reconsider the basic question we intended to
answer: What value does the market attach to an improvement in a product attribute? If there is only
one relevant customer, say individual i, then the
answer to the above question is
wik
:
wip

ð16Þ

This follows immediately from (14) with only one customer.4 Thus, at a conceptual level, wik and wip are core
parameters that should be part of any solution to the
original question. Yet, in a fully heterogeneous model,
where potentially all individuals are relevant for market performance, how individual parameters should
be weighted and aggregated is of primary interest.
A common practice in conjoint analysis for obtaining a single market value measure is to take a
weighted average of the individual ratios in (16)
(Wyner 1997). Following this approach with a sample
of N individuals we obtain
4

The same answer is obtained in conjoint analysis by considering
only Equation (11) and posing the condition that individual i’s
deterministic utility for alternative j stays unchanged. This can be
verified by dividing i’s preference for alternative j throughout by
the weight of price wip , as suggested by Srinivasan (1979). Thus,
utility is measured on a dollar-metric scale, and the coefficient for
attribute k becomes wik /wip .

403

OFEK AND SRINIVASAN
How Much Does the Market Value an Improvement in a Product Attribute?

!
N
1 X
wik
qi i :
Q i¼1
wp

ð17Þ

There are several important differences between the
expression in (17) and our MVAI measure (14). While
in (17), the summation is over individual ratios (wik /wip ),
in (14), we have a ratio of two summations. The numerator of (14) is an aggregate measure of sensitivity to
the level of attribute k, while the denominator is
an aggregate measure of sensitivity to price. From
a methodological perspective, the formulation in (14)
is more robust and less affected by outliers with extreme parameter values. Individuals with a very low
price weight (relative to their weight for attribute k)
would impact the overall measure suggested in (17)
far more than is called for. Such outliers will cause
the change in attribute k to be overevaluated. In particular, wip 5 0 for any one customer would make
(17) practically useless (unless this observation is
dropped). This problem is automatically avoided in
our MVAI formulation by separating the summations of customer attribute and price weights before
dividing.
While it is possible to reduce the outlier problem
by computing the (weighted) median of the set
fwik =wip gN instead of the weighted mean as in (17)
i¼1
(Orme 2001), both approaches would incorporate
only individual specific parameters (attribute and
price weights and individual quantity of purchase).
In contrast, an important aspect of the MVAI expression is that individual attribute and price weights
are multiplied by customer specific weights aij (see
(14)–(15)) prior to summation. In particular, two additional quantities, the probability of purchase hij
and the logit scale parameter li, now play a role in
determining how much each customer’s attribute
and price weights should contribute to the aggregate
measure. We discuss the significance of these factors
in turn.
3.4.2. The Impact of Probability of Purchase. The
setting for our analysis is that of a product category
in which each existing alternative has a specified

404

multiattribute location and price. The infinitesimal
changes we explore are taken about the current attribute values of the focal product. From Equation (12),
this implies that prior to any change, each individual has well-defined choice probabilities for the J
available alternatives. While these probabilities are
irrelevant when only one customer is analyzed (see
(16)), when the entire market is taken into account,
they are, in fact, important. Specifically, in (15), the
weight given to customer i has a probability-related
factor given by hij (1 2 hij ). This factor, arising in other
applications of the multinomial logit (e.g., Bucklin
et al. 1998), is a concave function of hij that attains
its maximum at hij 5 0.5 and approaches zero as
hij fi 0 or 1. What this means is that individuals who
either have a very low or a very high initial probability of choosing alternative j will bear far less on
the overall measure of market value than those with
a probability closer to 0.5. The intuition behind this
result is that the more extreme the probability of
purchase (either toward 0 or 1), the less likely
a change in product location or price will cause
a shift in choice probability or induce switching to
or from the focal product. For such individuals, the
original preference for the focal product is either
very low or very high, making them far less likely
to change their purchase probability. Thus, from the
perspective of impact on market share, we should
focus on those customers whose choice probabilities
exhibit maximal sensitivity. In the context of our
present model, these are the customers indifferent
between our product and the composition of all
others. Such customers are not too inclined or disinclined to choose the product so that their relative
impact is most relevant.5 It is noteworthy that even
when the number of products is J . 2, it is still true
that the most relevant customers to the firm making
5

We thank Professor Donald Lehmann for pointing out that our
result is analogous to political candidates exerting maximum effort on undecided voters, and direct marketers offering deals to
customers ‘‘on the fence.’’ The significance of marginal consumers,
for which a price increase in their consumed commodity induces
switching to another commodity, has also been pointed out by
Novshek and Sonnenshein (1979).

MARKETING SCIENCE/Vol. 21, No. 4, Fall 2002

OFEK AND SRINIVASAN
How Much Does the Market Value an Improvement in a Product Attribute?

the changes are those with hij fi 0.5, i.e., not those
with equal choice probabilities for all brands.6
As the term hij (1 2 hij ) is a function of all the alternatives in the market studied, the competitive structure of the product market is, in a sense, reflected.
In contrast, (17) depends only on the individual specific parameters. Because hij depends on j, we emphasize that with MVAI, the market’s value for the
same increment in a product attribute will differ, in
general, for each of the J products. MVAI would also
be different for the same focal product if we
changed the set of competing alternatives. In particular, we discuss the implications of increasing the
number of competitors in connection with the empirical application in §4.3.

3.4.3. The Role of the Logit Scale Parameters. In
addition to the hij (1 2 hij ) factor, the customer
weights aij are also affected by li. This term originates from the random utility component of the
multinomial logit model. A very small li (relative to
the weights wik ) indicates that the variance of the
random component is large, implying that the deterministic utility function has less impact on choices.
In fact, if li fi 0, Var(ei) fi ‘ and, consequently,
hij fi 1/(J 1 1) for j 5 0, 1, 2, . . ., J. That is, the deterministic utility function (and, hence, any attribute
change) has no effect on choice probabilities. On the
other hand, a very large value of li would mean that
customer i’s probability of choosing the product
with the highest tij approaches one. Therefore, the
presence of li allows for each customer’s contribution to the aggregate measure to be scaled to reflect
the relevance of the deterministic component of utility in establishing his or her choice probability.
6

Obviously, individual choice probabilities (hij ) tend to decrease as
the number of alternatives J increase. However, as this is true both
in the numerator and denominator of (14), even as J becomes very
large, MVAI and the average of individual ratios approach (17)
are generally expected to yield distinctly different values. See also
empirical results presented in §4.3.1.

MARKETING SCIENCE/Vol. 21, No. 4, Fall 2002

4. Empirical Illustration
We now illustrate how the proposed MVAI measure
can be estimated in the context of a new product development study. We discuss the advantages of our
proposed measure over alternative approaches and
allow for competitive price reactions.
4.1. Estimation of Multiattribute Preferences
The product category used in this particular study
was portable camera mounts. This represents a durable product category in which a typical customer
requires, at most, only one item, i.e., qi 5 1, "i. The
data are from 302 respondents who were contacted
as part of a graduate-level course in new product
design at Stanford University and who expressed interest in purchasing a product in the category.7 On
the basis of qualitative customer and retailer research, a set of five product attributes, in addition to
price, were predetermined as the most important
drivers of customer choice in the product category.
Participants were asked to rank, in order of likelihood of purchase, 18 full profile cards with all attributes having three possible values, thus avoiding
any number-of-attribute-levels effects (Wittink et al.
1990). (The ‘‘outside good’’ was not considered in
this application.) Using exploded logit (Ben-Akiva
et al. 1992, Chapman and Staelin 1982, Hausman and
Ruud 1987) we estimated the product of attribute
weights and logit scale parameters (liwik ) for each
attribute (including price) and for each individual.8
4.2.

Calculating Market Value for an Attribute
Improvement
To simulate a product market with competing alternatives, we used two existing commercial products
(UltraPod and Q-Pod) and a third designed by the
students (GorillaPod). See Table 1 for a description
of these three products (the Camera Critter and Half
Dome products are discussed in §4.3.1). Given the
7
See Srinivasan et al. (1997) for more details regarding the course
entitled ‘‘Integrated Design for Marketability and Manufacturing’’
(IDMM).
8
In the exploded logit model, it is not possible to estimate li and
wik separately; only their product can be estimated.

405

OFEK AND SRINIVASAN
How Much Does the Market Value an Improvement in a Product Attribute?

Table 1

Attribute Levels of Portable Camera Mount Products

Product
UltraPod
Q-Pod
GorillaPod
Camera Critter
Half Dome

Weight
(oz.)
2.0
3.5
4.6
1.7
5.7

Size

a

0.98
0.84
1.27
0.80
1.2

Set Up
Time (min.)
0.98
0.84
0.50
0.62
0.42

Stability
1.8
2.5
2.3
2.5
3.0

b

Flexibility

Table 2

Comparing Methods for Calculating Market Value for
a
an Attribute Improvement

c

1.96
2.17
2.84
1.8
2.33

a

Where 1 represents a camera mount that can fit in a standard
pocket, and 3 only in a standard book bag.
b
Where 1 means a camera mount stable enough under light-medium wind conditions for a small camera with a built-in lens, and 3
for a full-size camera with a large lens.
c
Where 1 means that the Positioning Flexibility of a camera mount
is low, and 3 is high. Positioning Flexibility is the degree to which
the product can be adapted to various terrains (flat-uneven) and
be adjusted for height (inches-feet) and angle (fixed-complete rotational freedom).

MVAI
Attribute (cost)e
Weight (0.49)
Size (0.23)
Set Up Time (1.41)
Stability (0.31)
Flexibility (0.26)
f

RMSE
g
MAD
h
MAPD

b

UltraPod Q-Pod GorillaPod
1.59
1.12
0.95
1.10
0.74

1.66
1.26
0.99
1.35
0.86

1.58
1.15
1.00
1.26
0.89

Ave.
c
Ratios

Med.
d
Ratios

2.87
2.06
2.06
2.58
1.93

0.60
0.03
0.68
0.63
;0

1.14
1.13
101.4%

0.84
0.78
68%

a

All values in the table are positive to reflect value of attribute
improvement (in tens of dollars).
P
b
i
i
MVAIjk 5 i hij (1 2 hij )(liwk )/Ri hij (1 2 hij )(liwp ).
c
i
i
Average of individual ratios 5 (1/N) Ri (wk /wp ), N 5 302.
d
i
i
Median of individual ratios from the set fwk =wp gN .
i¼1
e
Number in parentheses denotes the marginal cost of improving the
attribute (in tens of dollars).
f
Root mean squared deviation from MVAI (based on the 15 different
items, i.e., 3 products 3 5 attributes, for which the deviation from
MVAI can be computed).
g
Mean absolute deviation from MVAI.
h
Mean of absolute percent deviation from MVAI.

product market, it is straightforward to use the approach described in §3.3 to calculate a dollar MVAI
for each attribute. This is done with respect to all
five attributes and for each of the three products.
The resulting values are given in columns 2–4 of
Table 2. To highlight the benefits of our proposed
measure compared to other commonly used methods,
we computed the following two additional measures
based on the value each individual, when analyzed
separately, attaches to the attribute change (see (16)).
The average and median of these individual ratios
are reported in columns 5 and 6 of Table 2, respectively. It is clear from the table that both alternative
measures yield very different results compared to
those arising from the MVAI measure. In particular,
the values generated by the averaging approach differ by 101.4% from the MVAI measure (averaged
across all five attributes and all three products), and
are always higher (in some cases, by as much as
161% for the UltraPod product with respect to Flexibility). This is an indication of the upward bias re-

sulting from averaging individual measures because
of outliers (i.e., very small price weights) in the data
as discussed in §3.4.1.9 Also note that while Size and
Set Up Time have equal values with the averaging
approach, this is not the case with MVAI.
The values generated by selecting the median of
individual ratios differ from MVAI by 68%. While
the outlier problem is avoided with this approach,
the values are biased downward for all attributes.10
In addition, Set Up Time is the most highly valued
attribute with this approach, while it is only the
fourth highly valued attribute in terms of MVAI.
The difference between MVAI and the median is
most pronounced for Flexibility, with a near zero
median value. We also note that, while MVAI gener-

9
This was true even after constraining individual price weights so
that each individual had at least some minimal level of price senP
sitivity according to (j(liwip )Dpj)/( K11j(liwik )Dkj) > 1/t(K 1 1),
t 5 2, and Dk,p the feasible range of attribute k or price, respectively.
We also set t 5 1, t 5 4, and obtained qualitatively similar results.

10
The distribution of wik /wip is positively skewed due to the division by wip ; hence, the median is expected to be lower than the
mean. Note that in our empirical application, the median values
are not only lower than in the averaging approach but are also
lower than the corresponding MVAI values.

406

MARKETING SCIENCE/Vol. 21, No. 4, Fall 2002

OFEK AND SRINIVASAN
How Much Does the Market Value an Improvement in a Product Attribute?

Table 3

UltraPod Profitability from an Attribute Change When
Incorporating Competitive Price Reactions
b

d

Attribute Changed
Weight
Size
Set Up Time
Stability
Flexibility

MVAIa
Based
108.5
87.8
245.4
77.9
47.3

Competitive-Reaction Scenario

c

No Reaction
109.9
89.2
244.9
79.7
48.2

Price Reaction
67.3
69.8
267.0
40.5
21.3

a

Based on (5), we report Qmj (MVAI 2 MC), where MVAI and MC
(the marginal cost of improving the attribute) are as given in Table
2, Q 5 N 5 302, and mUltraPod 5 32.65% based on the initial product
market attribute levels (Table 1).
b
To be on a comparable scale to the MVAI-based values, we report
Dpj /Dxjk where Dpj is the difference in UltraPod profits before and
after the attribute change, and Dxjk is the amount by which the attribute was improved.
c
Given the attribute change by UltraPod, all firms simultaneously adjusted prices so that the market was in a Nash price equilibrium.
d
Each attribute was changed by 5% of its range of values.

ally produces different values for each of the alternative products, the average and median of individual
ratios do not depend on the product considered (see
(17)). For example, MVAI gives UltraPod a higher
market value for Size than Stability, while the reverse is true for Q-Pod and GorillaPod.
The derivation of MVAI in (3)–(5) shows that it is
meaningful to compare MVAI with the marginal cost
of improving each attribute. From prototypes of
several products the students actually built, we
obtained through multiple regression analysis, an
approximate marginal cost associated with each attribute improvement (@c/@xk). These marginal costs
are given in parentheses in column 1 of Table 2. Several interesting conclusions emerge. When subtracting the marginal cost increase associated with
attribute improvement from MVAI, Set Up Time is
clearly not profitable for any of the products. However, comparing this cost of improvement with the
value given by the averaging approach predicts that
Set Up Time is a profitable attribute to improve.
4.3. Incorporating Competitive Price Reactions
Our MVAI measure (and the corresponding comparison of it with marginal cost of attribute improve-

MARKETING SCIENCE/Vol. 21, No. 4, Fall 2002

ment (5)–(6)) was developed under the assumption
that competitors do not react to the attribute and
price changes made by the repositioning firm. In
many cases, it is more realistic to assume that competitors will, in fact, react by adjusting their own
prices. To explore this possibility, we allowed one
firm, UltraPod, to improve each of its five attributes
(one at a time by 5% of the range of allowable attribute values).11 Two possible scenarios were considered. In the No-Reaction Scenario, only UltraPod
modified a product attribute and then changed its
price to achieve maximal profits. In the Price-Reaction Scenario, we let all firms simultaneously change
prices subsequent to the attribute change by UltraPod and required that the market be in a Nash price
equilibrium (just as it was in such an equilibrium
prior to the UltraPod move).
The results of exploring the two scenarios are provided in Table 3. Column 2 of Table 3 gives the
MVAI-based criterion (5) for evaluating profitability
(using the marginal costs of improving attributes
given in parentheses in column 1 of Table 2). Columns 3–4 of Table 3 give UltraPod profitability
under the two Competitive-Reaction Scenarios described above on a per unit attribute changed, thus
allowing direct comparison with the MVAI-based
criterion. As expected, UltraPod profitability levels
for all attribute improvements decrease as one
moves from column 3 to 4.
The results demonstrate the relevance of MVAI.
First, note that the MVAI-based profitability measure
is closely related to the profitability values in the
No-Reaction Scenario.12 This corroborates the theoretical derivation of MVAI for small changes taken
about the existing product. Second, the attribute
changes for which MVAI is positive (negative) continue to be positive (negative) after taking competitive price reactions into account and are generally
the same rank order. The only exception is that Size
becomes slightly more profitable than Weight in the
11

We also explored 1% and 2.5% improvements and obtained similar results.
12
The MVAI result in column 2 of Table 3 is based on an infinitesimal change in attribute value, while column 3 is based on a 5%
change in attribute value.

407

OFEK AND SRINIVASAN
How Much Does the Market Value an Improvement in a Product Attribute?

Price-Reaction Scenario. This intuitively occurs because Weight is the most highly valued attribute for
all three competing products, as can be gleaned
from the MVAI values in Table 2. Hence, an UltraPod change in that attribute triggers a stronger pricing response than Size does. In the Appendix, we
provide a theoretic explanation for why comparison
of MVAI to the incremental cost of attribute improvement continues to be important when other
firms react through price to the focal product’s attribute change, as well as how profitability generally
decreases as competitors react with price changes.
4.3.1. Sensitivity of MVAI to Expanding the
Competitive Set. As noted in §3.4.2, the presence of the
term hij (1 2 hij ) in both the numerator and denominator
of (14) renders MVAI values for each product sensitive
to the set of competing alternatives. In particular, the introduction of additional alternatives can change the
MVAI values of existing products.13 The sign of these
changes depends on how the relative attractiveness of
each product’s attribute levels and equilibrium price are
affected by additional products in the set. Table 4 presents MVAI values when the set of alternatives is
increased from three to four and then to five products
(a description of the additional alternatives is given in
the last two rows of Table 1). As can be seen, when the
existing three alternatives are joined by Camera Critter,
all MVAI values for UltraPod decrease. This is because
the price charged for UltraPod decreases considerably;
hence, it tends to attract (on a probabilistic basis) individuals who are relatively price sensitive (with high wip ).
Given that the presence of Camera Critter also tends to
reduce the relative appeal of UltraPod product attributes,14 the denominator of (14) (reflecting price sensitivity) tends to increase relative to the numerator, resulting
in lower MVAI values for all attributes. For Q-Pod, the
situation is reversed. It is the highest-priced alternative

Table 4

MVAI as a Function of the Number of Competing
a
Products
b

MVAI
Products in
c
Competitive Set

Setup
Time Stability Flexibility

Weight

Size

1. UltraPod (8.84)
2. Q-Pod (9.89)
3. GorillaPod (9.53)

1.59
1.66
1.58

1.12
1.26
1.15

0.95
0.99
1.00

1.10
1.35
1.26

0.74
0.86
0.89

1.
2.
3.
4.

UltraPod (7.72)
Q-Pod (9.22)
GorillaPod (8.50)
Camera Critter (8.22)

1.54
1.72
1.55
1.63

1.09
1.33
1.14
1.20

0.92
1.01
1.01
0.99

1.04
1.44
1.26
1.25

0.73
0.94
0.95
0.76

1.
2.
3.
4.
5.

Ultra Pod (7.15)
Q-Pod (8.53)
GorillaPod (7.75)
Camera Critter (7.49)
Half Dome (10.39)

1.52
1.69
1.50
1.59
1.93

1.06
1.30
1.10
1.17
1.44

0.90
0.99
0.99
0.97
1.19

0.98
1.37
1.18
1.19
1.85

0.70
0.90
0.92
0.72
1.17

a

A description of the products is given in Table 1. All values in the
table are positive to reflect value of attribute improvement (in tens
of dollars).
P
P
b
i
i
MVAIjk 5 i hij (1 2 hij )(liwk )/ i hij (1 2 hij )(liwp ).
c
Number in parenthesis is the equilibrium price (in tens of dollars).

(with the margin of difference to the second- most expensive product having increased); hence, it tends to attract individuals who are less price sensitive. The
denominator thus decreases relative to the numerator,
offsetting the fact that Camera Critter is more attractive
on some attributes. For GorillaPod, the situation is
mixed. Its price is in a middle range; hence, on the attributes on which it dominates (Set Up Time and Flexibility), MVAI values tend to increase, while for
attributes the new alternative is more attractive on, they
tend to decrease. Similar considerations help explain
MVAI changes between the four- and five-product
scenarios. It is noteworthy that because the fifth alternative (Half Dome) is relatively highly priced, it now
tends to attract the price-insensitive individuals and,
consequently, MVAI values for Q-Pod all decrease.

13

We stress that individual attribute weights are estimated
through a conjoint study (rank ordering 18 hypothetical product
profiles) independently from the set of alternatives in the simulated product market.
14
The only exception is Flexibility, that is slightly higher for UltraPod (see Table 1) and, consequently, MVAI decreases the least for
this attribute (from 0.74 to 0.73).

408

5. Conclusion
This paper intended to theoretically derive a measure of the MVAI (in dollar terms). We achieved this

MARKETING SCIENCE/Vol. 21, No. 4, Fall 2002

OFEK AND SRINIVASAN
How Much Does the Market Value an Improvement in a Product Attribute?

goal by examining the change in demand for the
firm’s product as a result of an incremental improvement in a particular attribute, with its price
being adjusted optimally. We obtained a closed form
expression for MVAI by using the multinomial logit
framework, which has been extensively used to
model marketing phenomena. The expression for
MVAI reveals that the market’s valuation is not
a simple average of the individual customers’ valuations for an improvement in a product attribute.
While the trade-off between price and attribute level
needs to be captured by the corresponding price
and attribute weights, there are other factors to be
considered. We find that individual-level parameters
should be differentially weighted according to probability of purchase of the firm’s product. ‘‘Extreme’’
customers, i.e., those with very high or very low
probability of purchasing the focal product, are far
less relevant in determining the market value of an
attribute change because their probabilities are not
as sensitive to the proposed changes. How much
the market values an improvement in a particular
attribute depends on which competitive product is
asking the question. The measure developed scales
individual weights by a factor related to the inverse
of the variance of the random component of utility.
It should also be noted that attribute and price
weights are summed separately. This kind of formulation, as opposed to averaging the ratio of these individual weights, is less susceptible to the influence
of outliers.
Correctly assessing the market value for a product
attribute change has important implications for
firms engaging in product modifications. By comparing the market value with the marginal cost of
providing that change, the firm can determine
whether an attribute improvement would be profitable. Taking this comparison a step further, MVAI
allows the firm to establish, given the current market structure, which product characteristics are
most worthwhile to modify and where to direct
R&D efforts.
The customer weights in (14)–(15) suggest an approach for segmenting customers following the attribute change. In particular, the focal firm would want

MARKETING SCIENCE/Vol. 21, No. 4, Fall 2002

to target customers for whom both the customer
weight aij and the individual value (wik /wip ) are high.15
While the main purpose of this paper was to derive a theory-based measure for the market-level valuation of an attribute improvement, we also provided
an empirical application demonstrating how the proposed measure can be estimated in practice. The
results clearly highlight the benefit of the MVAI
measure in mitigating the effects of data outliers.
They also reveal the potential misleading implications of using alternative approaches (average or
median of individual values), which do not incorporate the relative appeal to customers of competing
products, in determining the market’s value for an
improvement (Orme 2001). The empirical analysis
also explored the possibility of competitive reactions
in prices by rivals. Even under this scenario, the
MVAI-based criterion for evaluating profitability
was found to be useful in providing directional insights to a firm considering product modifications.
There are several limitations in our study. First,
our measure is only applicable to attributes that are
differentiable in the neighborhood being analyzed;
thus, we are unable to treat discontinuous product
features, e.g., a car manufacturer contemplating the
addition of a side-impact air bag. However, such
discontinuous features can be analyzed by the conceptual framework of §2. Second, although the multinomial logit is widely accepted for modeling
marketing phenomena, our result (14) may not hold
precisely under other probabilistic choice models.
Third, in the competitive reaction analysis (§4.3 and
Appendix) we considered only price reactions. Suppose the focal firm improves attribute xk by y%. If
competitors match the focal firm’s attribute change,
and all firms optimally adjust prices, it can be
shown (under certain assumptions) that all equilibrium profits will remain the same as prior to the
attribute change. However, competitors may react
by changing multiple attributes by amounts that may
be different from y%. Recognizing this, the focal firm
15

For another customer segmentation approach that uses statistical
significance between differences in probability of purchase in an
industrial marketing context, see Gensch (1984).

409

OFEK AND SRINIVASAN
How Much Does the Market Value an Improvement in a Product Attribute?

may also change multiple attributes by differing
amounts. The equilibrium analysis under product and
pricing changes is a worthwhile subject for future
research. Finally, our empirical study used real
data gathered from individuals interested in making
a purchase in the camera mount category, and involved product design and development by students. It would be useful for future research to
validate the desirable properties of our MVAI measure (compared to the alternative approaches) in
other categories and using firm-level sales data prior
and subsequent to a product improvement.

Acknowledgments
The authors thank Miklos Sarvary, Muhamet Yildiz,
the Editor-in-Chief, the Area Editor, and two anonymous reviewers for their constructive comments.
The authors also acknowledge the support of
William Simpson and the Faculty Research Computing Center at the Harvard Business School.

dpj
@pj @pj dpà X @pj dpÃ
j
j9
¼
þ
þ
:
dxjk @xjk @pj dxjk j96¼ j @pj9 dxjk

ðA2Þ

From (5), (A1), and (A2), we obtain

 X
dpj
@mj =@xjk @cj
@pj dpÃ
j9
¼ Qmj À
À
:
þ
dxjk
@mj =@pj @xjk
@pj9 dxjk
j96¼j

ðA3Þ

The first term on the right-hand side of (A3) is known as the ‘‘direct effect’’ while the second term represents a ‘‘strategic effect’’
(see Tirole 1988, pp. 326 and 327). The direct effect reflects the impact of firm j’s own actions on its profits and is essentially the
same as that in Equation (5), clarifying that MVAI plays a crucial
role in assessing the profitability of attribute modifications even
when we require the market to be in a pricing equilibrium. The
strategic effect arises from the simultaneous reaction of all other
firms to the improvement firm j is administering. In the multinomial logit market share model, products compete as substitutes
(in prices), i.e.,
@pj/@pj9 . 0 (j 6¼ j9); hence, the sign of the strategic effect is largely determined by the equilibrium price adjustments of the rival
firms. If the direct effect is positive and outweighs the strategic
effect, the sign of the profit change will be positive. See the Technical Appendix for such an example.17

References
Appendix
Competitive Price Reactions
We now examine the case where all J firms reoptimize their prices
following an attribute change by a single firm and require that the
market be in a pricing equilibrium.16 In an interior solution, all J
firms must be simultaneously solving
@pj9
¼ 0;
@pj9

for j9 ¼ 1; 2; . . . ; J:

ðA1Þ

Once again, we examine the total effect on the profitability of the
focal brand j arising from an infinitesimal change dxjk
16
In many industry settings, price is typically a variable that firms
can change relatively quickly while other changes are harder to
administer in the short run due to production constraints. This assumption is common in empirical studies of differentiated products (e.g., Berry 1994). It is also similar to the one used by Horsky
and Nelson (1992) and Choi and DeSarbo (1994). While both studies use a conjoint type method for establishing demand, they do
not provide general analytical insights as to how changes in a particular attribute level affect equilibrium profits and how these effects depend on individual-level parameters. Choi et al. (1990)
provide some analytic comparative statics around the Nash price
equilibrium for the duopoly case but assume that customers are
homogeneous in both their sensitivity to price and quality.

410

Ansari, A., N. Economides, A. Ghosh. 1994. Competitive positioning in markets with nonuniform preferences. Marketing Sci.
13 (3) 248–273.
Ben-Akiva, M., S. R. Lerman. 1985. Discrete Choice Analysis: Theory
and Applications to Travel Demand. MIT Press, Cambridge,
MA.
———, T. Morikawa, F. Shiroishi. 1992. Analysis of the reliability
of preference ranking data. J. Bus. Res. 24 (2) 149–164.
Berry, S. T. 1994. Estimating discrete-choice models of product differentiation. RAND J. Econom. 25 (2) 242–262.
Bucklin, R. E., G. Russell, V. Srinivasan. 1998. A relationship
between market share elasticities and brand switching probabilities. J. Marketing Res. 35 (February) 99–113.
Chapman, R. G., R. Staelin. 1982. Exploiting rank ordered choice
set data within the stochastic utility model. J. Marketing Res.
19 (August) 288–301.
Choi, C., W. DeSarbo, 1994. A conjoint-based product designing
procedure incorporating price competition. J. Product Innovation Management 11 (5) 433–450.
———, ———, P. Harker. 1990. Product positioning under price
competition. Management Sci. 36 (2) 175–199.
Gensch, D. H. 1984. Targeting the switchable industrial customer.
Marketing Sci. 3 (1) 41–54.
17
The Technical Appendix is available at the INFORMS Marketing
Science website at http://www.informs.org.

MARKETING SCIENCE/Vol. 21, No. 4, Fall 2002

OFEK AND SRINIVASAN
How Much Does the Market Value an Improvement in a Product Attribute?

Green, P., V. Srinivasan. 1978. Conjoint analysis in consumer research: Issues and outlook. J. Consumer Res. 5 (September)
103–123.
———, ———. 1990. Conjoint analysis in marketing: New developments with implications for research and practice. J. Marketing 54 (October) 3–19.
Griffin, A. 1997. PDMA research on new product development
practices: Updating trends and benchmarking best practices.
J. Product Innovation Management 14 (6) 429–458.
Hauser, J. R., S. M. Shugan. 1983. Defensive marketing strategies.
Marketing Sci. 2 (4) 319–360.
Hausman, J. A., P. A. Ruud. 1987. Specifying and testing econometric models for rank-ordered data. J. Econom. 34 (Jan./
Feb.) 83–104.
Horsky, D., P. Nelson. 1992. New brand positioning and pricing in
an oligopolistic market. Marketing Sci. 11 (2) 133–153.
McFadden, D. 1974. Conditional logit analysis of qualitative
choice behavior. P. Zarembka, ed. Frontiers in Econometrics.
Academic Press, New York.
Novshek, W., H. Sonnenshein. 1979. Marginal consumers and neoclassical demand theory. J. Political Econom. 87 (6) 1368–1376.

Orme, B. 2001. Assessing the monetary value of attribute levels
with conjoint analysis: Warnings and suggestions. Sawtooth
Solutions Customer Newsletter (Spring) Sawtooth Software Inc.,
Sequim, WA.
Srinivasan, V. 1979. Network models for estimating brand-specific
effects in multi-attribute marketing models. Management Sci.
25 (January) 11–21.
———, W. S. Lovejoy, D. Beach. 1997. Integrated product design
for marketability and manufacturing. J. Marketing Res. 34
(February) 154–163.
Tirole, J. 1988. The Theory of Industrial Organization. MIT Press,
Cambridge, MA.
Wagner, H. M. 1975. Principles of Operations Research, 2nd ed. Prentice-Hall, Englewood Cliffs, NJ, 539.
Wittink, D. R., L. Krishnamurthi, D. J. Reibstein. 1990. The effect
of difference in the number of attribute levels on conjoint
results. Marketing Lett. 1 (June) 113–123.
Wyner, G. 1997. Personal communication, Mercer Management
Consulting, New York.

This paper was received June 21, 1999, and was with the authors 25 months for 3 revisions; processed by Dick Wittink.

MARKETING SCIENCE/Vol. 21, No. 4, Fall 2002

411