||Which of the two cars do you like more / do you prefer / is more attractive / is more impressive / is more
desirable / may bring you more satisfaction? Skoda or Mercedes?
||Which of the two cars will you buy more likely? Skoda or Mercedes?
This page gives a very simplified and, to some extent, inexact descriptions that might not pass a
rigorous inspection. Please take it as a way to a better understanding of a collection of terms dealing with
Preference and Utility. A good textbook is recommended.
Notion of utility
In consumer theory, utility of a product is a measure of the relative usefulness, desirability or satisfaction
from a (real or anticipated) purchase, owning or using the product. The utility measure depends on the way utility
is defined. If utility can get a value, it is called cardinal. If only an order can be known utility is called
It must be underlined that utility is not a property of the product. It is projection of the product properties
into the mind of its (potential) purchaser or user and reflects the attitudes, perceptions, needs, expectations,
satisfaction, current situation, etc., in respect to the product. From the point of MR, utility gets a value
reflecting the respondent's reaction to the product.
It is generally assumed that events on the market are driven by maximization of utility by decision-makers
(users, consumers, agents, ...). They make their decisions under economic (budget), social (habits) and
nominal (legal) restrictions, consumption (physiological or psychological) saturation constraints, etc. Other
conditions such as availability (distribution) and accessibility (existence) are influential. Note that the
description encompasses, but is not limited to, a rational consumer.
Direct and indirect utility
|Direct utility, as understood in microeconomics, is a
measure of a desirability of the product. It is independent from price and is non-decreasing with a quantity
of the product ("more is better").
- Direct utility is a measure without a reference (zero) value.
- Any monotonic transformation of direct utility gives utility describing the same product.
- In case of a linear transformation, the relative distances between utilities do not change.
- A nonlinear monotonic transformation of utilities does not change ordinal preferences of the
products. However, the relative distances between utilities will change which can have direct impact
on their interpretation.
- A transformation can be found so that the utility measure is an interval scaled variable.
|Provided direct utilities of products in a potential consumption bundle and
the behavioral model (composed utility function) are known, the simplest approach to explanation of expected
behavior is to choose one or more appropriate constraints. There are two basic approaches.
- The total direct utility is maximized with a budget constrained from above.
- The total cost is minimized for a (total) utility constrained from below.
|Both approaches lead to an identical result called indirect utility
of the consumption bundle. This is known as "econometric dualism".
- An important result under a constraint is so called corner solution suggesting either no or constant demand
of a product. It is independent from price in some price interval, i.e. price elasticity is equal to zero in
the close neighborhood of the corner point.
A disadvantage of the direct utility is that it tells little about a possible purchase of the product it
represents if the adequate model of demand is unknown. Another view of utility has been searched for. Based on the
purchased bundle of products, WARP
- Weak Axiom of Revealed Preference was postulated by Paul Samuelson (1947):
A bundle X is said to be weakly revealed preferred to a bundle Y if the latter could have
been purchased when the former was.
By the "could have been" is meant that the cost of (not purchased) bundle Y was not exceeding the cost of the
bundle X at the time of the purchase. "Weakly preferred" means that the preference of the bundle X was at least
equal or higher than that of the bundle Y.
Based on the above axiom completed with transitivity (leading to SARP - Strong Axiom of Revealed Preferences),
the ordinal (order-based) preference utility of the bundle can be defined. Instead of thinking about individuals
having utility functions, we may think of individuals as having preference orderings. An important aspect of the
revealed preferences definition is the implicit inclusion of prices of the products in the bundle under naturally
existing constraints. This is a view completely different from the indirect utility that must have been obtained
from the total direct utility by its maximization under artificially introduced constraints. On the other hand, no
judgment can be made about products that were not purchased and that would make the cost of the bundle higher than
that of the purchased bundle X. Direct utilities of such products might be both lower or higher than of those
present in the bundle X.
- Revealed preferences have proved to be very useful in building macro-economic consumer indexes.
- While the measure of the direct utility is based on an elusive "desirability" or "satisfaction", it might
seem that a measure of preference utility could be based on the money spent for the product in a consumption
bundle. Unfortunately, this is not possible. An increase in income nearly always leads to a higher diversity
of products in a consumption bundle compared to the increase of quantity of already consumed products. While
money is strictly additive, utilities of products and their quantities are not (cf. marginal properties,
diminishing utility and goods diversity postulates in the literature dealing with utility).
- Experimental estimation of direct or preference utilities for individual products from known consumption
bundles is not feasible. One would need to know at least as many consumption bundles as there are products.
This is why it is common to partition consumed goods into categories and treat each product category
separately as an aggregated item. Consumer segmentation is a must since preferences are expected to be
different in different segments.
- The notion of ordinal preferences makes basis for some marketing research methods. BPTO (Brand-price
Trade-off) and SIMALTO (Simultaneous Multi-attribute Trade-off, developed by the Xerox Corp. in the 50's) are
the well known. BPTO has significant bias and has been replaced by CBC
- Choice Based Conjoint. SIMALTO may be seen as a precursor of today's BYO - Build Your Own (product,
Virtually every product in a consumption bundle has been selected and chosen by an individual. Choosing one
actual bundle over another, consumers convey important information about their tastes. A problem arises in the
attempt to estimate utility of products absent from a consumption bundle of an individual (or a segment of
individuals) in the revealed preferences approach. As it cannot be distinguished between a low utility product
from an unaffordable, unavailable or inaccessible one, a direct or preference utility cannot be determined for
such a product.
In order to overcome the inconvenience with the "missing" products, a probabilistic approach to presence of a
product in the consumption bundle has been adopted. The probabilistic utility is expressed in terms of probability
of the product choice leading to the presence of the product in the consumption bundle. It is defined [McFadden,
1974] as (natural) logarithm of the probability of the product incidence in the consumption bundle. The resulting
dimensionless unit is logit. The utility is essentially indirect and its definition has some important
- The choice probability of a product is dependent on the choice set while the direct utility is not supposed to
be. A behavioral model that converts utility into a choice probability must be known. Evidently, the model and
its appropriateness to the observed behavior are crucial.
- Utility of a product can be determined only as a difference from utility of a reference product. This is in
agreement with a general understanding of cardinal utility being an interval scaled measure with an arbitrarily
- A choice set that leads to no choice is a valid one. The event can be interpreted as a choice of a product
from a generalized group of products assigned the term "outer goods".
Understanding probabilistic utility
Models of probabilistic human behavior are the essence of DCM-based experiments, analysis of the observed data
and projection of the results onto the market. Understanding probabilistic utilities can be based on their
relationship with both indirect and preference utilities. To clarify this aspect we will fall back to the weak
axiom of preferences.
- It is useful to decompose the whole consumption bundle into categories of products that are possible
substitutes and to consider only one category.
- If the category is treated independently, the influence of some constraints, namely that of the budget,
can be relaxed. The expenditure limits, be it "mental" or real affordability, will become part of the
- Other limits, such as physiological or need limits, both lower or upper, may apply. The behavioral model
should respect them.
- The bundle of all "consumed" products in the category can be considered as a result of choices made by
individuals from some choice sets.
- The real choice sets are given mainly by the existing distribution channels and exposition of the goods.
These factors may be used for segmentation and weighting of sets and choices obtained from a simulated
- From the above is clear that a universal choice set cannot be simulated. Its coarse approximation may be
the set of all commonly available products in the category. Accepting this, availability (distribution
channels), exposition (outlet size), accessibility (outlet location), etc., will not be included in the
- It is presumed that choices of the products found in the consumed bundle were made by individuals rather than
by consumers from certain predetermined segments as is usual in classical econometric applications. When
utilities on an individual basis are determined, segments based on personal preferences (tastes) can be
Now, the weak axiom of revealed preference can be changed to a form explaining (rather than defining) the
A product X is said to be preferred in probability to a product Y if
the latter could have been chosen when the former was.
By "could have been" is meant that the product Y was present in the choice set. There are no other requirements.
The "weakly preferred" term of the WARP axiom has been replaced by the "in probability" term. By the
"probability" is meant that the choice is subject to a random error. This assumption is the base of RUT - Random utility Theory originally
proposed by Thurstone (1927) as a way to understand and model choices between pairs of stimuli.
- The term "probabilistic utility" is preferred to "random utility" on these pages. A set of utilities is
often a result of a research and the word "random" have aroused suspicions ("Why would we want to have random
Comparison of utility types
There are important similarities and differences between direct utilities, indirect utilities, revealed
preferences and probabilistic utilities.
- Direct utility
- Direct utility is cardinal (has value). It does not accommodate for the influence of any constraints, e.g.
- If stripped of affordability, availability, accessibility, social impropriety, etc., i.e. leaving "a pure
desire", direct utility is sometimes called felicity.
- Direct utility is well suited for modeling influences of various constraints such as budget limits while
this is hardly possible with probabilistic utilities or preferences.
- Indirect utility
- Utility obtained by direct utility or cost optimization while imposing social, psychological, physical and
other constraints on consumption.
- A formulation of indirect utility to accommodate the influence of price requires an exact formulation of
- Preferences are essentially ordinal.
- Revealed preferences concern a bundle of goods rather than individual products.
- The influence of price on individual products is implicit. Utilities of individual products cannot be
determined. (However, marginal utilities can be determined).
- While estimation of revealed preferences in population or its segments is quite common, it is impossible
for an individual.
- Revealed preferences are useful for tracking in time (cf. welfare indexes).
- Preferences have sound theoretical fundamentals. Nevertheless, due to their inherently ordinal character,
they have a limited use in demand prediction.
- Probabilistic utility
- Probabilistic utility is cardinal (has value) and may concern a single product or a bundle of items.
- The influence of price is implicit. Provided the behavioral model is known the influence can be
- Probabilistic utilities can be estimated from market or research data, and both these groups of values can
be compared and matched.
- Probabilistic utilities, in contrast to revealed preferences, can be assessed for individuals.
- Probabilistic utility for an individual should be viewed as a conditional "as if" utility reflecting the
implicit personal (endogenous) constraints, exogenous (market, society, income, stereotypes) influences and
the conditions at which they were obtained (home, studio, field interview).
- Models for neo-classical econometric demand prediction can be matched with probabilistic models. In this
way, a probabilistic utility can be expressed in a form of an indirect utility.
- The meaning of a probabilistic utility, when stripped off of the price term (and possibly of other
influences leading to a saturation), can be related to a direct utility.
From an analysis of a DCM experiment, the obtained probabilistic utilities can be related to average indirect
utilities of products provided only one product from the choice set has been selected. The indirect utility of a
set of products can be obtained as expected indirect utility. This approach is useful in estimating acceptability
of product portfolios, bundles, packages, etc.
Decomposing probabilistic utility as a sum of terms of which just one is function of price, as required by the conjoint additive kernel, is theoretically possible only in some
trivial cases. The function is generally non-linear, but can be linearized or approximated as linear under certain
- Cardinalists claim utility can be measured. Ordinalists claim utility cannot be measured. The reader may
choose. If such a choice is believed to have an estimable probability, the reader gets on the claim of