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WORKING PAPER

ALFRED P. SLOAN SCHOOL OF MANAGEMENT

Dewey

MASS. INST. Tt'JH.

MAY 2 2 75

DEV-'EY U3RARY

A UTILITY MODEL FOR PRODUCT POSITIONING

Ralph L. Keeney

International Institute for Applied Systems Analysis

Laxenburg, Austria

Gary L. Lilien

Massachusetts Institute of Technology

Cambridge, Massachusetts

March 1975

WP #777-75

MASSACHUSETTS

INSTITUTE OF TECHNOLOGY

50 MEMORIAL DRIVE

CAMBRIDGE, MASSACHUSETTS 02139

A UTILITY MODEL FOR PRODUCT POSITIONlr.'G

Ralph L. Kecney

International Institute for Applied Systems Analysis

Laxenburg, Austria

Gary L. Lilian

Massachusetts Institute of Technology

Cambridge, Massachusetts

March 1975

WP #777-/5

s^^

V^?*"^

V?ii^^

ri

â– 75

Ot

^â€¢i^

ViB^^^

MAY 2 2 1975

â– :ECElvÂ£D 1

ABSTRACT

A model and procedure are proposed to help design and position consumer

durable and industrial products. These products are characterized on the

one hand by little data of repeat-purchase variety but on the other by a high

level of consistency between product preference and purchase behavior. The

procedure is based on utility theoretic concepts for assessing preference

and inferring probable behavior. Several numerical examples are included.

0724765

-3-

Firms continually face positioning and design issues related to their

products. The product design issue is essentially: "What physical and

psychological characteristics would I like my (new or revamped) product to

have? [to maximize profit, market share, or more generally, the firm's ex-

pected utility]." This product design question is relevant to both new and

existing products. The characteristics that can be controlled include price,

several dimensions of use or quality, packaging, flavor (perhaps), etc. The

firm generally has explicit control over physical quantities (price, taste,

objective performance) and implicit control, through message and communica-

tions design, over the psychological quantities (a young-swinging beer, e.g.).

How should products be designed and how can those designs be changed

during a product's life cycle to meet a firm's objectives? There have been

two modes of analysis to help answer these questions. One approach is that

of the psychometricians. In particular, making use of similarity data, Stefflre

[13] developed "perceptual maps" of market structures. He suggested introduc-

ing brands in areas of the map which were relatively vacant. Other work has

followed; a summary of the multidimensional scaling literature is available

in Green and Carmone [4] and Green and Rao [5] .

Another approach to the design question is provided by marketing model

builders. Many attempts have been made to establish functional forms which

relate product attributes, such as price, to output measures such as market

share. Kotler [10] provides a review of much of the literature.

A more recent, promising approach is one taken by Urban [14]. Urban

uses psychometric techniques to map the space of consumers' attitudes toward

existing brands as well as an "ideal" brand. He postulates that the farther

a brand is from the ideal point, the lower its market share should be. The

-4-

procedure, v/hich models the trial and repeat-purchase processes separately,

assumes that a brand's long run market share is a parameterized quadratic

function of its distance from the ideal point. That function is then cali-

brated using actual market share data from the market. The m.odel was shown

to predict ultimate market share for a number of new brands quite well. But,

due to data requirements, the model's use would seem to be lim.ited to frequent-

ly purchased products. Virtually all work in these areas has been limited to

frequently purchased products, mainly due to the greater availability of data.

Thus, the problem of analytically positioning consumer durable products and

most industrial products has been largely ignored.

A main difference between consumers purchasing packaged foods, and com-

panies purchasing minicomputers is that industrial purchasing behavior con-

forms more closely to prescribed criteria (see Webster and Wind [15]). Thus

one may "prefer" Krispy Krakers to "Whole Oats" and purchase Whole Oats any-

way due to "variety seeking," availability, or other random behavior. The

same inconsistency is not likely to occur in the purchase of a new minicom-

puter for one's firm or in the selection by parents of a medical treatment

for a baby's congenital defect.

The methodology suggested is designed to be used for precisely those

situations in which little data (of a repeat-purchase variety) are likely

to be available, but where the customers (individuals or firms) purchase

consistently with stated or inferred preferences. The approach is based on

utility theory (see Raiffa [H] for a discussion of the basic concepts of

utility theory) . It assumes customers have a von Neumann-Morgenstern utility

function defined over the product variables - that is, customers are expected

utility maximizers and "act as they should." Utility theory has been used in

the past mainly in a normative or prescriptive sense â€” telling decision

-5-

makors what they should do in given circumstances. The market situations con-

sidered here are, by definition, those in which individuals do what they should.

Hauser [6] is currently developing a structure for models of choice between

finite alternatives. His structure of the analytical process of choice includes:

(1) observation of behavior and measurement; (2) reduction and abstraction â€”

reducing the n'ar.iber of product dimensions to a few, "independent" ones and

labelling them; (3) compaction, developing brand preference measures; (4)

probability of choice, relating preference to purchase behavior, and (5) aggre-

gation, transforming probability of purchase measures to market share measures.

The methods developed here suggest augmenting the observation step, by

measuring attitudes in face-to-face interviews, through methods of direct

utility function assessment as in [1] , [8] , [9] . The compaction operation then

uses utility theory and the assumption of a von Neumann-Morgenstern utility

function obviates the probability-of -choice step. Aggregation is performed by

taking explicit account of consumer heterogeneity throughout the procedure.

The paper is organized as follows: Section 1 presents the formal struc-

ture of the model and introduces notation. Section 2 develops a general frame-

work of analysis which is applied to a simple, hypothetical example in Section 3.

1. The Model and Notation

Consider a well-defined product class with N firms F, , ..., F in a single,

^ IN

specific market. Let m. be the market share of F., and assume that each firm

X 1

makes a single product.

Let the set of attributes X , X , ... completely characterize a product,

where X could be price, X could be reliability ratings, etc. These attributes

would be attained by factor analysis of a series of well-defined product ratings

-6-

or perhaps non-metric scaling procedures, given a set of similarity judgments.

Both methods have proven effective though neither has established "superiority."

(Green, [3]) The output of these procedures, then, would be a reduced set of

product characteristics, X, , . . . X . A snecific level of X . is x . so a product

' 1' J - 3D

is completely described by x = (x , ... x ) . The product of firm F. will

be

denoted by x = (x , . . . , x ) . A no-product purchase x = (x , . . . x )

1 J 1 u

could be included for completeness.

Customers will be designated by C . . . , C . It v/ill be assuined that each

^ *^ 1 K

customer C has a von Neumann-Morgenstern utility function u (x/A) , where C 's

utility function is specified by the set of parameters X = (X ,...,A ). Assuming

each customer buys a product, utility theory suggests he should buy the product

of firm F. such that his utility is maximized.

1

Since viewing the problem from the firm's point of view will require the

same methodology regardless of the firm, let us take the viewpoint of firm F .

Firm F has certain objectives which could include maximizing market share,

maximizing profit, and so on. We postulcite that the objective function of F

is also specified by a von Neumann-Morgenstern utility function v over market

share m , profit and/or other variables.

There are uncertainties here for both firms and customers. The firm wants

to know utility functions for all customers in the market. This information about

customer heterogeneity will be expressed in the form of a probability distribu-

tion P, (M over the parameters A_ describing a random customer's utility function.

Thus, customers' utility functions are likely to differ so >_ does not take on

a single value, but, rather, is expressed as a probability distribution. Any

firm will not have perfect knowledge about the "true" P, (^) and will, in general,

attempt to estimate the distribution, entailing some error. Thus, we might con-

-7-

sider parameterizing the distribution to give P , (X/9) where 9 = (0 ,...,S )

X/0 â€” 1 T

indicates the uncertainties of F about the true X. We quantify that uncertain-

ty by the probability distribution P (9) .

6 â€”

Consumers in general will differ in their knov/ledge or attitude about the

characteristics X of firm F.'s product. From F 's perspective, this hetero-

geneity can be described by the parameterized probability distributions

P {x / ), i=l,...,N, where = ( r '^ r â– â€¢ â€¢ i ) .

To summarize here, our model contains (from F 's view):

1. An objective function v , which is a utility function, kno'.vn with cer-

tainty.

2. A distribution of utility functions u (x/A) which vary across the

heterogeneous customer population quantified by i^ (A/G) . The proba-

A/o â€” â€”

bility distribution Pn(9) quantifies this uncertainty.

6 â€”

3. A set of distributions of product-perceptions, P^ (x /i ), i=l,...,N,

also varying across the heterogeneous customer-population. The proba-

bility distribution P ((f) quantifies this uncertainty.

General Model Structure

In this section, we first consider the decision an individual consumer

must make and how his decisions are inputs to the decision-making processes

of the firm. Then we focus on how firms can use the model for product posi-

tioning decisions.

-8-

2. 1 Consumer Decisions

The consumer must decide v;hich if any of the products in the market to

buy given that he will buy at most one. Thus we explicitly consider the case

of a consumer not purchasing any product. Another possibility would have been

to define consumers as those who will buy one product and then formally include

uncertainty about the number of consumers in the market.

Our model does not explicitly include individual consumer uncertainty

about product characteristics. Rather, F. is uncertain both about the set

1

of consumer utility functions and the set of consumer product perceptions.

Explicit inclusion of consumer uncertainty would needlessly complicate the

croblem. We will assume that C has a utility function u, (x) and his choices

k k â€”

are not to buy a product and receive x or to buy the product of F. and re-

ceive x , i=l,2,...N. He will choose the option x* giving him the highest

utility where x* is defined by

(1) u (x*) = max {Uj^(2i^) >

i=0,l, . . .N

In the case of uncertainty, the consumer should choose the product of

ik

firm F. which maximizes his expected utility. If P (x) represents the judg-

i X

ment of consun\er C about firm F.'s jjroduct, the option x_* should be chosen

such that

E[u, (x*)] = max Ju, (x) P^ (x ) dx ,

'^- i=0,l N '^ ^ -

where E [u (x*) ] is the expected utility of the product of firm F. (where F

k â€” 1 o

designates the no product option) for individual C .

-9-

2. 2 Firm Decisions Under Certainty

Under certainty, firm F should maximize its market utility v . Here we

assume the distribution of utility functions is known and that customers do

not vary in their perception of brand characteristics (that is 9 and ^ , i=l,

2,...,N, are known). The condition of certainty could be used as a first cut,

as less inform.ation is required here to attain a product design decision. Firm

F has a product with characteristics x and the population utility functions

are represented by u (x/^) where P (X) represents the population heterogeneity.

From (1) it follows that a consumer with parameters A_ will choose firm Ft ' s

product if

(2) \^^/h) > \^^/h) ' h=0,l,..., N, h?^l,

where we neglect ties in terms of equally desireable alternatives. The pro-

portion of consumers m, who choose x is

(3) m^ = /p. {X)d\_,

where At is that set of A's such that (2) holds. Neglecting ties, A , A, ,...,A^,

1 o 1 N

are mutually exclusive and collectively exhaustive so N

y m. = 1 as reauired.

''1

i=0

1 -a

Suppose F IS considering changing its product position from x to 5< .

Then a new m can be determined exactly as m was using (2) and (3) .

Suppose a new firm with utility function v is trying to enter a volume

inelastic market with a maximum profit product (x) ; that is, say,

V = m â€¢ G â€¢ (s - C(x) ) - d(x) where s is unit product selling price, C is

the unit cost of the product (x) , G is the market volume, m is its market

-10-

share, and d(x) is the fixed development cost (plant, R&D, etc.) associated

with (x) . Tlien v is the profit associated v;ith the new product. There will

likely be a set Q of alternate product positions, generically denoted by x,

that the new firm could attain. Given the existing products in the market,

for each possible x there is a set A(x) (perhaps null) defined by

(4)

A(x) E {A: u, (x/X) > u (x /X) , i=0, 1, . . . , N}

â€” k â€” k â€” â€”

The best decision for this firm is to choose x in Q to maximize

(5)v(x) =

/ P (A_)dA

XcA(x)

_J

(G â€¢ (s-C(x)

d(x)

where the first terra in brackets in (5) represents the market share of product (x)

and v(x) is the profit associated with the product. Under "nice" conditions

it may be possible to simply differentiate v(x) with respect to x to determine

the product position to maximize profit.

2.3 Firm Decisions Uncer Uncertainty

The problems under uncertainty are parallel but more complicated than

those with certainty. The two sources of uncertainty are the firms' imper-

fect knowledge about preference (or utility) heterogeneity, characterized by 9_,

and imperfect knowledge about perceptual heterogeneity, characterized by

^, i=0,l,...,N.

We v/ill assume here, for simplicity, that perceptual heterogeneity and

utility heterogeneity are independent within individuals. Fix the utility

heterogeneity parameters as A_ and select a customer with utility function

u{x/A) at random. That customer's selection will be from the set of product

-11-

characteristic distributions P (^1 /^ ) Â» i=0,l,...,N. The product purchased

should be x* such that

E(u(x*/_X,^) = max E(u(x )A,

ALFRED P. SLOAN SCHOOL OF MANAGEMENT

Dewey

MASS. INST. Tt'JH.

MAY 2 2 75

DEV-'EY U3RARY

A UTILITY MODEL FOR PRODUCT POSITIONING

Ralph L. Keeney

International Institute for Applied Systems Analysis

Laxenburg, Austria

Gary L. Lilien

Massachusetts Institute of Technology

Cambridge, Massachusetts

March 1975

WP #777-75

MASSACHUSETTS

INSTITUTE OF TECHNOLOGY

50 MEMORIAL DRIVE

CAMBRIDGE, MASSACHUSETTS 02139

A UTILITY MODEL FOR PRODUCT POSITIONlr.'G

Ralph L. Kecney

International Institute for Applied Systems Analysis

Laxenburg, Austria

Gary L. Lilian

Massachusetts Institute of Technology

Cambridge, Massachusetts

March 1975

WP #777-/5

s^^

V^?*"^

V?ii^^

ri

â– 75

Ot

^â€¢i^

ViB^^^

MAY 2 2 1975

â– :ECElvÂ£D 1

ABSTRACT

A model and procedure are proposed to help design and position consumer

durable and industrial products. These products are characterized on the

one hand by little data of repeat-purchase variety but on the other by a high

level of consistency between product preference and purchase behavior. The

procedure is based on utility theoretic concepts for assessing preference

and inferring probable behavior. Several numerical examples are included.

0724765

-3-

Firms continually face positioning and design issues related to their

products. The product design issue is essentially: "What physical and

psychological characteristics would I like my (new or revamped) product to

have? [to maximize profit, market share, or more generally, the firm's ex-

pected utility]." This product design question is relevant to both new and

existing products. The characteristics that can be controlled include price,

several dimensions of use or quality, packaging, flavor (perhaps), etc. The

firm generally has explicit control over physical quantities (price, taste,

objective performance) and implicit control, through message and communica-

tions design, over the psychological quantities (a young-swinging beer, e.g.).

How should products be designed and how can those designs be changed

during a product's life cycle to meet a firm's objectives? There have been

two modes of analysis to help answer these questions. One approach is that

of the psychometricians. In particular, making use of similarity data, Stefflre

[13] developed "perceptual maps" of market structures. He suggested introduc-

ing brands in areas of the map which were relatively vacant. Other work has

followed; a summary of the multidimensional scaling literature is available

in Green and Carmone [4] and Green and Rao [5] .

Another approach to the design question is provided by marketing model

builders. Many attempts have been made to establish functional forms which

relate product attributes, such as price, to output measures such as market

share. Kotler [10] provides a review of much of the literature.

A more recent, promising approach is one taken by Urban [14]. Urban

uses psychometric techniques to map the space of consumers' attitudes toward

existing brands as well as an "ideal" brand. He postulates that the farther

a brand is from the ideal point, the lower its market share should be. The

-4-

procedure, v/hich models the trial and repeat-purchase processes separately,

assumes that a brand's long run market share is a parameterized quadratic

function of its distance from the ideal point. That function is then cali-

brated using actual market share data from the market. The m.odel was shown

to predict ultimate market share for a number of new brands quite well. But,

due to data requirements, the model's use would seem to be lim.ited to frequent-

ly purchased products. Virtually all work in these areas has been limited to

frequently purchased products, mainly due to the greater availability of data.

Thus, the problem of analytically positioning consumer durable products and

most industrial products has been largely ignored.

A main difference between consumers purchasing packaged foods, and com-

panies purchasing minicomputers is that industrial purchasing behavior con-

forms more closely to prescribed criteria (see Webster and Wind [15]). Thus

one may "prefer" Krispy Krakers to "Whole Oats" and purchase Whole Oats any-

way due to "variety seeking," availability, or other random behavior. The

same inconsistency is not likely to occur in the purchase of a new minicom-

puter for one's firm or in the selection by parents of a medical treatment

for a baby's congenital defect.

The methodology suggested is designed to be used for precisely those

situations in which little data (of a repeat-purchase variety) are likely

to be available, but where the customers (individuals or firms) purchase

consistently with stated or inferred preferences. The approach is based on

utility theory (see Raiffa [H] for a discussion of the basic concepts of

utility theory) . It assumes customers have a von Neumann-Morgenstern utility

function defined over the product variables - that is, customers are expected

utility maximizers and "act as they should." Utility theory has been used in

the past mainly in a normative or prescriptive sense â€” telling decision

-5-

makors what they should do in given circumstances. The market situations con-

sidered here are, by definition, those in which individuals do what they should.

Hauser [6] is currently developing a structure for models of choice between

finite alternatives. His structure of the analytical process of choice includes:

(1) observation of behavior and measurement; (2) reduction and abstraction â€”

reducing the n'ar.iber of product dimensions to a few, "independent" ones and

labelling them; (3) compaction, developing brand preference measures; (4)

probability of choice, relating preference to purchase behavior, and (5) aggre-

gation, transforming probability of purchase measures to market share measures.

The methods developed here suggest augmenting the observation step, by

measuring attitudes in face-to-face interviews, through methods of direct

utility function assessment as in [1] , [8] , [9] . The compaction operation then

uses utility theory and the assumption of a von Neumann-Morgenstern utility

function obviates the probability-of -choice step. Aggregation is performed by

taking explicit account of consumer heterogeneity throughout the procedure.

The paper is organized as follows: Section 1 presents the formal struc-

ture of the model and introduces notation. Section 2 develops a general frame-

work of analysis which is applied to a simple, hypothetical example in Section 3.

1. The Model and Notation

Consider a well-defined product class with N firms F, , ..., F in a single,

^ IN

specific market. Let m. be the market share of F., and assume that each firm

X 1

makes a single product.

Let the set of attributes X , X , ... completely characterize a product,

where X could be price, X could be reliability ratings, etc. These attributes

would be attained by factor analysis of a series of well-defined product ratings

-6-

or perhaps non-metric scaling procedures, given a set of similarity judgments.

Both methods have proven effective though neither has established "superiority."

(Green, [3]) The output of these procedures, then, would be a reduced set of

product characteristics, X, , . . . X . A snecific level of X . is x . so a product

' 1' J - 3D

is completely described by x = (x , ... x ) . The product of firm F. will

be

denoted by x = (x , . . . , x ) . A no-product purchase x = (x , . . . x )

1 J 1 u

could be included for completeness.

Customers will be designated by C . . . , C . It v/ill be assuined that each

^ *^ 1 K

customer C has a von Neumann-Morgenstern utility function u (x/A) , where C 's

utility function is specified by the set of parameters X = (X ,...,A ). Assuming

each customer buys a product, utility theory suggests he should buy the product

of firm F. such that his utility is maximized.

1

Since viewing the problem from the firm's point of view will require the

same methodology regardless of the firm, let us take the viewpoint of firm F .

Firm F has certain objectives which could include maximizing market share,

maximizing profit, and so on. We postulcite that the objective function of F

is also specified by a von Neumann-Morgenstern utility function v over market

share m , profit and/or other variables.

There are uncertainties here for both firms and customers. The firm wants

to know utility functions for all customers in the market. This information about

customer heterogeneity will be expressed in the form of a probability distribu-

tion P, (M over the parameters A_ describing a random customer's utility function.

Thus, customers' utility functions are likely to differ so >_ does not take on

a single value, but, rather, is expressed as a probability distribution. Any

firm will not have perfect knowledge about the "true" P, (^) and will, in general,

attempt to estimate the distribution, entailing some error. Thus, we might con-

-7-

sider parameterizing the distribution to give P , (X/9) where 9 = (0 ,...,S )

X/0 â€” 1 T

indicates the uncertainties of F about the true X. We quantify that uncertain-

ty by the probability distribution P (9) .

6 â€”

Consumers in general will differ in their knov/ledge or attitude about the

characteristics X of firm F.'s product. From F 's perspective, this hetero-

geneity can be described by the parameterized probability distributions

P {x / ), i=l,...,N, where = ( r '^ r â– â€¢ â€¢ i ) .

To summarize here, our model contains (from F 's view):

1. An objective function v , which is a utility function, kno'.vn with cer-

tainty.

2. A distribution of utility functions u (x/A) which vary across the

heterogeneous customer population quantified by i^ (A/G) . The proba-

A/o â€” â€”

bility distribution Pn(9) quantifies this uncertainty.

6 â€”

3. A set of distributions of product-perceptions, P^ (x /i ), i=l,...,N,

also varying across the heterogeneous customer-population. The proba-

bility distribution P ((f) quantifies this uncertainty.

General Model Structure

In this section, we first consider the decision an individual consumer

must make and how his decisions are inputs to the decision-making processes

of the firm. Then we focus on how firms can use the model for product posi-

tioning decisions.

-8-

2. 1 Consumer Decisions

The consumer must decide v;hich if any of the products in the market to

buy given that he will buy at most one. Thus we explicitly consider the case

of a consumer not purchasing any product. Another possibility would have been

to define consumers as those who will buy one product and then formally include

uncertainty about the number of consumers in the market.

Our model does not explicitly include individual consumer uncertainty

about product characteristics. Rather, F. is uncertain both about the set

1

of consumer utility functions and the set of consumer product perceptions.

Explicit inclusion of consumer uncertainty would needlessly complicate the

croblem. We will assume that C has a utility function u, (x) and his choices

k k â€”

are not to buy a product and receive x or to buy the product of F. and re-

ceive x , i=l,2,...N. He will choose the option x* giving him the highest

utility where x* is defined by

(1) u (x*) = max {Uj^(2i^) >

i=0,l, . . .N

In the case of uncertainty, the consumer should choose the product of

ik

firm F. which maximizes his expected utility. If P (x) represents the judg-

i X

ment of consun\er C about firm F.'s jjroduct, the option x_* should be chosen

such that

E[u, (x*)] = max Ju, (x) P^ (x ) dx ,

'^- i=0,l N '^ ^ -

where E [u (x*) ] is the expected utility of the product of firm F. (where F

k â€” 1 o

designates the no product option) for individual C .

-9-

2. 2 Firm Decisions Under Certainty

Under certainty, firm F should maximize its market utility v . Here we

assume the distribution of utility functions is known and that customers do

not vary in their perception of brand characteristics (that is 9 and ^ , i=l,

2,...,N, are known). The condition of certainty could be used as a first cut,

as less inform.ation is required here to attain a product design decision. Firm

F has a product with characteristics x and the population utility functions

are represented by u (x/^) where P (X) represents the population heterogeneity.

From (1) it follows that a consumer with parameters A_ will choose firm Ft ' s

product if

(2) \^^/h) > \^^/h) ' h=0,l,..., N, h?^l,

where we neglect ties in terms of equally desireable alternatives. The pro-

portion of consumers m, who choose x is

(3) m^ = /p. {X)d\_,

where At is that set of A's such that (2) holds. Neglecting ties, A , A, ,...,A^,

1 o 1 N

are mutually exclusive and collectively exhaustive so N

y m. = 1 as reauired.

''1

i=0

1 -a

Suppose F IS considering changing its product position from x to 5< .

Then a new m can be determined exactly as m was using (2) and (3) .

Suppose a new firm with utility function v is trying to enter a volume

inelastic market with a maximum profit product (x) ; that is, say,

V = m â€¢ G â€¢ (s - C(x) ) - d(x) where s is unit product selling price, C is

the unit cost of the product (x) , G is the market volume, m is its market

-10-

share, and d(x) is the fixed development cost (plant, R&D, etc.) associated

with (x) . Tlien v is the profit associated v;ith the new product. There will

likely be a set Q of alternate product positions, generically denoted by x,

that the new firm could attain. Given the existing products in the market,

for each possible x there is a set A(x) (perhaps null) defined by

(4)

A(x) E {A: u, (x/X) > u (x /X) , i=0, 1, . . . , N}

â€” k â€” k â€” â€”

The best decision for this firm is to choose x in Q to maximize

(5)v(x) =

/ P (A_)dA

XcA(x)

_J

(G â€¢ (s-C(x)

d(x)

where the first terra in brackets in (5) represents the market share of product (x)

and v(x) is the profit associated with the product. Under "nice" conditions

it may be possible to simply differentiate v(x) with respect to x to determine

the product position to maximize profit.

2.3 Firm Decisions Uncer Uncertainty

The problems under uncertainty are parallel but more complicated than

those with certainty. The two sources of uncertainty are the firms' imper-

fect knowledge about preference (or utility) heterogeneity, characterized by 9_,

and imperfect knowledge about perceptual heterogeneity, characterized by

^, i=0,l,...,N.

We v/ill assume here, for simplicity, that perceptual heterogeneity and

utility heterogeneity are independent within individuals. Fix the utility

heterogeneity parameters as A_ and select a customer with utility function

u{x/A) at random. That customer's selection will be from the set of product

-11-

characteristic distributions P (^1 /^ ) Â» i=0,l,...,N. The product purchased

should be x* such that

E(u(x*/_X,^) = max E(u(x )A,

1 2

Online Library → Ralph L. Keeney → A utility model for product positioning → online text (page 1 of 2)