Utility Representation and Strictly Increasing Transformations (Ordinal Utility)

The set of bundles is the space of consumption pairs for two goods/services, denoted as X.
In the transcript, a typical bundle is x = (x1, x2) with each component nonnegative: $x1 \in \mathbb{R}+, \quad x2 \in \mathbb{R}+.$
One way to think about X is as the set of all bundles with nonnegative components; there is a mention that in some contexts consumption could be negative, but for most examples we take bundles as nonnegative.
A specific common representation is: $X = {(x1, x2): x1 \ge 0, x2 \ge 0} = \mathbb{R}_+^2.$
A bundle y is another element of this same set, so y = (y1, y2) with the same nonnegativity constraints.
For intuition, x can be viewed as a bundle of two goods (e.g., group 1 and group 2, or service 1 and service 2).

Preferences over bundles can be represented by a utility function u: X → \mathbb{R}.
The transcript uses the relation: $u(x) \text{ is weakly greater than } u(y) \iff \text{ (bundle } x \text{ is at least as preferred as } y).$
When this condition holds for all x, y, we say that the utility function u represents the preferences.
In symbols, a representation satisfies: $x \succeq y \quad \text{iff} \quad u(x) \ge u(y).$
Practical interpretation: Instead of directly comparing bundles via the preference relation, we can compare the numbers given by the utility function; a higher utility value means a preferred bundle.

The transcript emphasizes that a utility function representing preferences does not have a unique numerical scale.
If you transform the utilities by a strictly increasing function, the ranking/order of bundles is preserved, hence the same preferences are represented.
This means there are infinitely many utility functions that can represent the same preferences.

A function f: \mathbb{R} → \mathbb{R} is strictly increasing if for all a,b with a > b, we have f(a) > f(b).
Such functions are sometimes also described as strictly monotone increasing (the course also uses the term monotone, but strictly increasing is the precise term here).
Graphically, a strictly increasing function has a graph that rises as the input rises; it cannot have flat spots where the output does not increase with the input.
The transcript notes that monotone increasing can include non-strict cases (nondecreasing), but for the purpose here we require strictly increasing.

Consider a base utility function u: X → \mathbb{R} that represents preferences.
A transformation b is called a strictly increasing transformation of u if there exists a strictly increasing function f: \mathbb{R} → \mathbb{R} such that for all x ∈ X,
$\, b(x) = f(u(x)).$
In words: b is a new utility representation obtained by applying a strictly increasing transformation to the original utility values.
The function f is the transformation; it maps the original utility values to a new set of numbers, while preserving the ranking of bundles.
For a given x, you first evaluate u(x) to obtain a real number, then apply f to get b(x).
This is why two different utility functions can represent the same preferences if they are related by a strictly increasing transformation.

Because f is strictly increasing, the ordering of utilities is preserved: for any x, y, if u(x) > u(y) then b(x) = f(u(x)) > f(u(y)) = b(y).
Therefore, b represents the same preferences as u; the only thing that changes is the numerical scale, not the ranking.
This leads to the concept of ordinal utility: what matters for preferences is the order, not the exact magnitudes of the utility numbers.
The transcript notes that under this view, the utility function is not unique; many different functions can represent the same ranking.
A generic way to write this relationship is:
$\exists \; f: \mathbb{R} \to \mathbb{R} \text{ strictly increasing such that } b(x) = f(u(x)) \quad \forall x \in X.$

The transcript hints at upcoming examples (e.g., perfect substitutes) where different utilites can represent the same preferences.
Common illustration: for perfect substitutes, a typical base utility is linear in goods, e.g., u(x) = a x1 + b x2, \quad a,b > 0. Then, applying any strictly increasing f (such as exponential, square, or a positive affine transformation) yields another representation, preserving the same preference ordering.
The key takeaway is not the exact numbers, but the order they induce over bundles.

Bundle space: $X = {(x1, x2) : x1 \ge 0, x2 \ge 0} = \mathbb{R}_+^2.$
Utility representation: x ≽ y iff $u(x) \ge u(y).$
Non-uniqueness: There exist infinitely many utility functions that represent the same preferences; only the ranking matters.
Strictly increasing transformation: A function f: \mathbb{R} → \mathbb{R} with a > b \implies f(a) > f(b).
Transformation of utility: If there exists a strictly increasing f s.t. $b(x) = f(u(x))$ for all x, then b represents the same preferences as u.

Ordinal utility vs cardinal utility: The passage highlights that only the order of preference matters for ordinal utility, not the absolute magnitudes of the numbers.
Policy and decision making: Because many representations exist, economists focus on the rank-order properties of choices rather than specific numerical values.
Practical modeling: When calibrating models, any strictly increasing transformation of a baseline utility is acceptable; this flexibility can be used to simplify calculations or align with data scales, as long as the ranking is preserved.

Always specify the bundle space and the domain of x, y: $x, y \in X = \mathbb{R}_+^2.$
Remember the core definition: $x \succeq y \iff u(x) \ge u(y).$
If you apply a strictly increasing transformation, the new function still represents the same preferences: $\exists \; f \text{ strictly increasing with } b(x) = f(u(x)).$
Small but important distinction: Strictly increasing vs non-decreasing; only the strictly increasing case preserves strict ranking in all cases used here.
Expect examples that illustrate how different representations (due to transformations) yield identical choice predictions, especially in the case of perfect substitutes and similar preference structures.