Ranking Bundles

Once the three axioms are in place, consumers can rank every possible bundle of goods from most preferred to least preferred. But how do we represent that ranking in a model? Enter utility — a numerical scoring system that assigns a number to each bundle so that higher-numbered bundles are preferred.

Imagine you’re rating Netflix shows. You might give Breaking Bad a 10, The Office a 9, and a random cooking show a 4. The numbers don’t mean Breaking Bad is “2.5 times better” than the cooking show — they just encode ranking. Utility works the same way. If $U(A)=50$ and $U(B)=30$, all we know is that $A$ is preferred to $B$. The difference (20) has no inherent meaning. This is called ordinal utility — only the ordering matters, not the size of the gaps.

One key consequence: any monotonic transformation of a utility function represents the same preferences. If $U(x)=x^2$ ranks bundles correctly, so does $V(x)=2x^2+7$, or $W(x)=\ln (x^2)$. They all preserve the same ordering. This is why utility is ordinal — infinitely many different functions can encode the same preferences.