# 3.2: Utility Theory

- Page ID
- 24480

Learning Objectives

- In this section we discuss economists’ utility theory.
- You will learn about assumptions that underlie individual preferences, which can then be mapped onto a utility “function,” reflecting the satisfaction level associated with individuals’ preferences.
- Further, we will explore how individuals maximize utility (or satisfaction).

**Utility theory** bases its beliefs upon
individuals’ preferences. It is a theory postulated in economics to
explain behavior of individuals based on the premise people can
consistently rank order their choices depending upon their
preferences. Each individual will show different preferences, which
appear to be hard-wired within each individual. We can thus state
that individuals’ preferences are intrinsic. Any theory, which
proposes to capture preferences, is, by necessity, abstraction
based on certain assumptions. Utility theory is a **positive
theory** that seeks to explain the individuals’ observed
behavior and choices.The distinction between normative and positive
aspects of a theory is very important in the discipline of
economics. Some people argue that economic theories should be
normative, which means they should be prescriptive and tell people
what to do. Others argue, often successfully, that economic
theories are designed to be explanations of observed behavior of
agents in the market, hence positive in that sense. This contrasts
with a **normative theory**, one that dictates that
people should behave in the manner prescribed by it. Instead, it is
only since the theory itself is positive, after observing the
choices that individuals make, we can draw inferences about their
preferences. When we place certain restrictions on those
preferences, we can represent them analytically using a
**utility function**—a mathematical formulation that
ranks the preferences of the individual in terms of satisfaction
different consumption bundles provide. Thus, under the assumptions
of utility theory, we can assume that people behaved as if they had
a utility function and acted according to it. Therefore, the fact
that a person does not know his/her utility function, or even
denies its existence, does not contradict the theory. Economists
have used experiments to decipher individuals’ utility functions
and the behavior that underlies individuals’ utility.

To begin, assume that an individual faces a set of consumption
“bundles.” We assume that individuals have clear preferences that
enable them to “rank order” all bundles based on desirability, that
is, the level of satisfaction each bundle shall provide to each
individual. This rank ordering based on preferences tells us the
theory itself has **ordinal utility**—it is designed
to study relative satisfaction levels. As we noted earlier,
absolute satisfaction depends upon conditions; thus, the theory by
default cannot have **cardinal utility**, or utility
that can represent the absolute level of satisfaction. To make this
theory concrete, imagine that consumption bundles comprise food and
clothing for a week in all different combinations, that is, food
for half a week, clothing for half a week, and all other possible
combinations.

The utility theory then makes the following assumptions:

- Completeness: Individuals can rank order all possible bundles.
Rank ordering implies that the theory assumes that, no matter how
many combinations of consumption bundles are placed in front of the
individual, each individual can always rank them in some order
based on preferences. This, in turn, means that individuals can
somehow compare any bundle with any other bundle and rank them in
order of the satisfaction each bundle provides. So in our example,
half a week of food and clothing can be compared to one week of
food alone, one week of clothing alone, or any such combination.
Mathematically, this property wherein an individual’s preferences
enable him or her to compare any given bundle with any other bundle
is called the
**completeness**property of preferences. - More-is-better: Assume an individual prefers consumption of
bundle A of goods to bundle B. Then he is offered another bundle,
which contains more of everything in bundle A, that is, the new
bundle is represented by αA where α = 1. The more-is-better
assumption says that individuals prefer αA to A, which in turn is
preferred to B, but also A itself. For our example, if one week of
food is preferred to one week of clothing, then two weeks of food
is a preferred package to one week of food. Mathematically, the
more-is-better assumption is called the
**monotonicity assumption**on preferences. One can always argue that this assumption breaks down frequently. It is not difficult to imagine that a person whose stomach is full would turn down additional food. However, this situation is easily resolved. Suppose the individual is given the option of disposing of the additional food to another person or charity of his or her choice. In this case, the person will still prefer more food even if he or she has eaten enough. Thus under the monotonicity assumption, a hidden property allows costless disposal of excess quantities of any bundle. - Mix-is-better: Suppose an individual is indifferent to the
choice between one week of clothing alone and one week of food.
Thus, either choice by itself is not preferred over the other. The
**“mix-is-better” assumption**about preferences says that a mix of the two, say half-week of food mixed with half-week of clothing, will be preferred to both stand-alone choices. Thus, a glass of milk mixed with Milo (Nestlè’s drink mix), will be preferred to milk or Milo alone. The mix-is-better assumption is called the “convexity” assumption on preferences, that is, preferences are convex. - Rationality: This is the most important and controversial
assumption that underlies all of utility theory. Under the
assumption of
**rationality**, individuals’ preferences avoid any kind of circularity; that is, if bundle A is preferred to B, and bundle B is preferred to C, then A is also preferred to C. Under no circumstances will the individual prefer C to A. You can likely see why this assumption is controversial. It assumes that the innate preferences (rank orderings of bundles of goods) are fixed, regardless of the context and time.

If one thinks of preference orderings as comparative relationships, then it becomes simpler to construct examples where this assumption is violated. So, in “beats”—as in A beat B in college football. These are relationships that are easy to see. For example, if University of Florida beats Ohio State, and Ohio State beats Georgia Tech, it does not mean that Florida beats Georgia Tech. Despite the restrictive nature of the assumption, it is a critical one. In mathematics, it is called the assumption of transitivity of preferences.

Whenever these four assumptions are satisfied, then the
preferences of the individual can be represented by a
**well-behaved utility function**.The assumption of
convexity of preferences is not required for a utility function
representation of an individual’s preferences to exist. But it is
necessary if we want that function to be well behaved. Note that
the assumptions lead to “a” function, not “the” function.
Therefore, the way that individuals represent preferences under a
particular utility function may not be unique. Well-behaved utility
functions explain why any comparison of individual people’s utility
functions may be a futile exercise (and the notion of cardinal
utility misleading). Nonetheless, utility functions are valuable
tools for representing the preferences of an individual, provided
the four assumptions stated above are satisfied. For the remainder
of the chapter we will assume that preferences of any individual
can always be represented by a well-behaved utility function. As we
mentioned earlier, well-behaved utility depends upon the amount of
wealth the person owns.

Utility theory rests upon the idea that people behave as if they make decisions by assigning imaginary utility values to the original monetary values. The decision maker sees different levels of monetary values, translates these values into different, hypothetical terms (“utils”), processes the decision in utility terms (not in wealth terms), and translates the result back to monetary terms. So while we observe inputs to and results of the decision in monetary terms, the decision itself is made in utility terms. And given that utility denotes levels of satisfaction, individuals behave as if they maximize the utility, not the level of observed dollar amounts.

While this may seem counterintuitive, let’s look at an example that will enable us to appreciate this distinction better. More importantly, it demonstrates why utility maximization, rather than wealth maximization, is a viable objective. The example is called the “St. Petersburg paradox.” But before we turn to that example, we need to review some preliminaries of uncertainty: probability and statistics.

## Key Takeaways

- In economics, utility theory governs individual decision making. The student must understand an intuitive explanation for the assumptions: completeness, monotonicity, mix-is-better, and rationality (also called transitivity).
- Finally, students should be able to discuss and distinguish between the various assumptions underlying the utility function.

## Discussion Questions

- Utility theory is a preference-based approach that provides a rank ordering of choices. Explain this statement.
- List and describe in your own words the four axioms/assumptions that lead to the existence of a utility function.
- What is a “util” and what does it measure?