In finite choice sets, we have the following notation:
\(X = \mathbb{R}^n_+\) be finite choices
\(c: P(X) \to X\) be the choice function
\(\succsim\) be the preference relation
\(U: \mathbb{R}^n_+ \to \mathbb{R}\) be the corresponding utility function.
What if Choices Become Infinite?
Different Levels of Finiteness
Finite
There exists a bijection \(X \leftrightarrow \{1, ..., n\}\).
each \(x \in X\) is paired with exactly one number in \(\{1, ..., n\}\)
each number in \(\{1, ..., n\}\) is paired with exactly one \(x \in X\)
nobody is left unpaired
Countrably Infinite
There exists a bijection \(X \leftrightarrow \mathbb{N} = \{1, 2, ...\}\).
Below are number sets by their finiteness:
\(\mathbb{Z} = \{..., -1, 0, 1, ...\}\) is countably infinite
\(\mathbb{Q} = \{a/b: a \in \mathbb{Z}, b \in \mathbb{N}\}\) is countably infinite
\(\mathbb{R}\) is not countably infinite
Countable Infinite
Proposition
\(\succsim\) be a rational preference and \(X\) be a countable set \(\Rightarrow\) there exists \(U: X \to \mathbb{R}\) representing \(\succsim\).
Proof of Proposition
Consider:
a countable choice set: \(X = \{x_1, ..., x_n, ...\}\).
a weight indicator: \(d_j = \frac{1}{2^j}\).
We would have \[\begin{aligned}
\Sigma_{j=1}^{\infty} d_j = \Sigma_{j=1}^{\infty} \frac{1}{2^j} = \lim_{n \to \infty} [\Sigma_{j = 1}^n \frac{1}{2^j}] = 1
\end{aligned}\]
We can define the utility function \(U: X \to \mathbb{R}\) as \[\begin{aligned}
U(z) = \Sigma_{\{j \in \mathbb{N}: (z, x_j) \in \succsim\}} d_j
\end{aligned}\]\(\blacksquare\)
Incountable Infinite
It is not always true that there exists a utility function \(U: X \to \mathbb{R}\) to represent a rational \(\succsim\).
the consumer would always prefer the bundle where the quantity of good \(A\) is more than it in any other bundles;
if two bundles have the same quantity of good \(A\), the consumer would always prefer the bundle where the quantity of good \(B\) is more than it in the other bundle.
We would need some other properties that \(\succsim\) must satisfy to make the utility function possible (namely continuity).
Continuous Preference Relation
Definition: Continuity
For all pairs \(x,y \in X\) with \(x \succ y\), we can always find \(\epsilon > 0\) such that:
\(|x - x'| < \epsilon\)
\(|y - y'| < \epsilon\)
implies \(x' \succ y'\).
Intuition
A strictly preferred pair should remain their preference status even under a sligtest change in any direction. We would not expect a rational person to freak out if a slightest change takes place.
Utility Representation
Proposition
If \(\succsim\) is complete, transitive, and continuous, then there exists a continuous \(U: \mathbb{R}^n_+ \to \mathbb{R}\) that represents \(\succsim\).