Choice theory

Introduction

The first goal of this page is to formalize the idea of preference using the concepts of preference relations (\(\succeq\)) and utility functions (\(u(\cdot)\)). Both are the cornerstones of economic theory, though utility functions are easier to work with and appear more frequently in applications. Since they are representations of the same underlying idea, we could use "preference" to refer to either concept.

Note that the following statements are equivalent ways to express the same preference:

1. Lucy (\(l\)) likes oatmeal (\(OAT\)) better than yogurt parfait (\(YOG\)).

2. \(OAT \succ_l YOG\)

   Read as: Lucy prefers oatmeal to yogurt parfait.

   Read as: Oatmeal is preferred to yogurt parfait (for Lucy).

3. \(u_l(OAT) > u_l(YOG)\)

   Read as: Lucy assigns higher utility to oatmeal than to yogurt parfait.

   Read as: Oatmeal has higher utility than yogurt parfait (for Lucy).

Also, we define some properties of preferences, and discuss how they can be formally represented using preference relations and utility functions. As shows, these properties introduce additional structure into our formal representation, which greatly simplifies analysis. This simplification comes at the cost of realism: some properties are safe to assume in many economic contexts, while others only apply in more limited situations. Therefore, we also need to learn how to interpret these properties to develop a clear sense of when such assumptions are appropriate.

Consider the following statement:

More leisure time is always preferred to less leisure time.

If we let the utility \(u(\cdot)\) be a function of the amount of daily leisure time \(T\), then we can establish a mathematical representation of this statement:

\(u(\cdot)\) is a strictly increasing function of \(T\).

As the title "choice theory" suggests, our ultimate concern is the choice-making process where economic agents make decisions in real-world scenarios.

To connect preference and choice making, we assume that agents choose options according to their preferences. Specifically, they choose options they "weakly prefer to all other available options," or equivalently, options that "have the highest utility." Every model of choice is characterized by such a choice rule that determines what choice(s) an agent should make. Note that preferences can exist independently of choice rules — such as stated preferences people report in questionnaires — but agents might act against these preferences when making actual choices.

This framework combines theory and data in two complementary ways. First, we use choice models to predict how agents behave. Second, we use observed choice data to test our models (when they generate testable predictions) or to inform model construction by imposing constraints on preference relations or calibrating utility functions. We call preferences inferred from this second exercise revealed preferences.

As such, our analysis focuses on how preference relations and utility functions must behave to remain consistent with real-world choice data. In addition, while the preference-utility framework is flexible enough to accommodate most observed behavior, we also explore alternative conceptual frameworks that can explain choice-making processes.

Preferences \(\succeq\)

We first formalize preferences using binary relations.

Let \(X\) be the the opporunity set, defined as the set of options available to an agent.

A preference \(\succ\) is a binary relation on \(X\).

A complete preference is one

Utility representation of preferences

TBD

Choice rules and revealed preference

TBD

Properties of preferences

TBD