Q1. 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?
A1. ........................................................
Q2. Which of the two cars will you buy more likely? Skoda or Mercedes?
A2. ........................................................

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 ordinal.

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". 
As aside


Preference utility

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.

As aside


Probabilistic utility

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 consequences.


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.

  1. It is useful to decompose the whole consumption bundle into categories of products that are possible substitutes and to consider only one category.
  2. The bundle of all "consumed" products in the category can be considered as a result of choices made by individuals from some choice sets.
  3. 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 identified. 

Now, the weak axiom of revealed preference can be changed to a form explaining (rather than defining) the probabilistic utility:

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.

As aside


Comparison of utility types

There are important similarities and differences between direct utilities, indirect utilities, revealed preferences and probabilistic utilities.

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 assumptions.

As aside