# How to Derive a Utility Function

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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.

## Utility: Concepts

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

## Basis for Utility Function Equations

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.

## Utility Function Examples

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 (​x1) and another that is simply linear (​x2). 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).

## Utility Function Calculator

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.