Question

JR Exercise 1.14: Suppose the preferences are represented by a continuous utility func- tion. Show that = satisfies Axioms 1, 2, and 3.

Answer 1

The standard axioms are completeness (given any two options x and y then either x is at least as good as y or y is at least as good as x), transitivity (if x is at least as good as y and y is at least as good as z, then x is at least as good as z), and reflexivity (x is at least as good as x).

Basic Representation Theorem
Suppose an agent chooses from a set of goods X = {a, b, c, . . .}. For example, one can think of
these goods as different TV sets or cars.
Given two goods, x and y, the agent weakly prefers x over y if x is at least as good as y. To
avoid us having to write “weakly prefers” repeatedly, we simply write x < y. We now put some
basic structure on the agent’s preferences by adopting two axioms.1
Completeness Axiom: For every pair x, y ∈ X, either x < y, y < x, or both.

Transitivity Axiom: For every triple x, y, z ∈ X, if x < y and y < z then x < z.
An agent has complete preferences if she can compare any two objects. An agent has transitive
preferences if her preferences are internally consistent. Let’s consider some examples.
First, suppose that, given any two cars, the agent prefers the faster one. These preferences are
complete: given any two cars x and y, then either x is faster, y is faster or they have the same
speed. These preferences are also transitive: if x is faster than y and y is faster than z, then x
is faster than z.
Second, suppose that, given any two cars, the agent prefers x to y if it is both faster and bigger.
These preferences are transitive: if x is faster and bigger than y and y is faster and bigger than
z, then x is faster and bigger than z. However, these preferences are not complete: an SUV
is bigger and slower than a BMW, so it is unclear which the agent prefers. The completeness
axiom says these preferences are unreasonable: after examining the SUV and BMW, the agent
will have a preference between the two.