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.

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)


\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.

Related Articles

How to Calculate the Wronskian
What Is a Counterexample in Algebra?
What Are Parametric and Nonparametric Tests?
Ways to Tell if Something Is a Function
The Difference Between Linear & Nonlinear Equations
Characteristics of a Linear Programming Problem
How to Find the Slope of a Nonlinear Line
Similarities of Univariate & Multivariate Statistical...
How to Calculate the Slope of a Line of Best Fit
How to Know if Something Is Significant Using SPSS
How to Find the Inverse of a Function
How to Interpret an Independent T Test in SPSS
How to Calculate a P-Value
How to Calculate Half of a Parabolic Curve
How to Find Y Value for the Slope of a Line
How is Calculus Used in Economics?
The Difference Between Sequence & Function
How to Find the Function in Math
What is the Definition of a Function Table in Math?
How to Interpret a Scatter Plot

Dont Go!

We Have More Great Sciencing Articles!