4.8 Modeling Preferences with Utility Functions

Economists are human, and humans are fundamentally lazy. If they have a tool that works well for one purpose, they often use it for others as well.

In the first part of the twentieth century, economists like Cobb and Douglas, Leontief, and others had modeled production processes using production functions. They tried to find production functions whose mathematical properties matched the physical production processes they were trying to model: for example, Leontief production functions were used for processes that used inputs in a specific ratio.

While economists use a wide range of utility functions, we’ll be interested in two main classes of functional forms in this course.

Utility functions for analyzing complements and substitutes

One of the key aspects of a utility function is how it describes the relationship between the goods it’s modeling:

• Complements are goods which an agents wants to consume together, like peanut butter and jelly, sugar and tea, or tennis balls and tennis racquets.
• Substitutes are goods which can fulfill the same purpose, like different flavors of jelly, different kinds of tea, or different brands of tennis balls.
• Independent goods are neither complements nor substitutes. For example, tea and tennis balls have no obvious relationship: they’re not used together, nor can one be used in place of the other.

While many different utility functions can model these phenomena, there are three special cases of this function that we’ll use a lot in this class: perfect complements, perfect substitutes, and the Cobb-Douglas utility function.

Quasilinear utility functions

Utility functions for analyzing situations in which one good has diminishing marginal utility, while another has constant marginal utility. These are called quasilinear utility functions, and will be particularly important when analyzing market decisions in Part II of this book.