How To Derive A Utility Function

In economics, a ​utility function​ represents a summation of an individual agent's (i.e., person's) formal ​preferences​. Those preferences, in any individual, are assumed to adhere to certain rules. For example, one of those rules is that given set of objects ​x​ and ​y​, one of the two statements "​x​ is at least as good as ​y​" and "​y​ is at least as good as ​x​" must be true in this context.

The language of preferences, translated into symbols, looks like this:

•​**​x​ > ​y​:**​ ​x​ is preferred ​strictly​ to ​y​
•​**​x​ ~ ​y​:**​ ​x​ and ​y​ are ​equally​ preferred
•​**​x​ ≥ ​y​:**​ ​x​ is preferred ​at least as much as​ is ​y​  

Relationships between utility, preferences and other variables can be used to derive utility functions and other useful equations in the area of decision-making.

Economists are interested in utility because it offers a mathematical framework upon which to model people's likelihood of making certain choices. Obviously, the goal of any marketing campaign is to increase the sales of a product. But if product sales rise or fall, it is important to understand cause and effect rather than simply observe a correlation.

Preferences have the property of ​transitivity​. This means that if x is at least as preferred as ​y​, and ​y​ is at least as preferred as ​z​, then ​x​ is at least as preferred as ​z​:

\(x ≥ y \text{ and } y ≥ z → x ≥ z\)

Although it seems trivial, they also have the property of reflexivity, meaning any group of objects ​x​ is always at least as preferred as itself:

\(x ≥ x\)

Not all preference relations can be expressed as a utility function. But if a preference relation is transitive, reflexive and continuous, then it can be expressed as ​continuous utility function​. Continuity here means that small changes to the set of objects does not greatly change the overall preference level.

A utility function ​U​(​x​) represents a true preference relation if and only if the preference and utility relationships are the same for all ​x​ in the set. That is, ​it must be true that​

\(\text{if } x_1≥ x_2 \text{ then } U(x_1) ≥ U(x_2)\)

that

\(\text{if } x_1 ≤ x_2 \text{ then } U(x_1) ≤ U(x_2)\)

and that

\(\text{if } x_1 \backsim x_2 \text{ then } U(x_1) \backsim U(x_2)\)

Note also that utility is ordinal, not multiplicative. That is, it is based on rank. That means that if ​U​(​x​) = 8 and ​U​(​y​) = 4, then ​x​ is strictly preferred to ​y​, because 8 is always higher than 4. But it is not "twice as preferred" in any mathematical sense.

Any utility function that has the form

\(U(x_1, x_2) = f(x_1) + x_2\)

has one "regular" component that is usually exponential in nature (​x​₁) and another that is simply linear (​x​₂). It is thus called a ​quasi-linear utility function​.

Similarly, any utility function that has the form

\(U(x_1, x_2) = x_1^ax_2^b\)

where ​a​ and ​b​ are constants greater that zero is called a ​Cobb-Douglas function​. These curves are hyperbolic, meaning that they come close to both the ​x​-axis and the ​y​-axis on a graph, but without touching either one, and are convex (bowed outward) in the direction of the origin (0, 0).

Online utility maximization calculators are available for finding any utility maximization graph as long as you have the raw data available. See Resources for an example.