An indifference map can be summarized by a single function. A utility function $U(x, y)$ gives each bundle a number so that preferred bundles get bigger numbers: $U(A) > U(B)$ exactly when $A \succ B$, and $U(A) = U(B)$ exactly when $A \sim B$. Its level curves are the Indifference Curves.
Key idea
Utility is ordinal: only the ranking it gives matters, not the size of the numbers. If $f$ is any strictly increasing function, $V = f(U)$ describes exactly the same preferences, as with Composition of Functions: feed $U$'s output into $f$.
Why it works
For an increasing $f$, $U(A) > U(B)$ if and only if $f(U(A)) > f(U(B))$, and equal inputs give equal outputs. So $V = f(U)$ ranks every pair of bundles the same way as $U$. Its level curves are the same curves with new labels: the curve $U = 6$ is the curve $V = f(6)$. A decreasing $f$ (such as $f(u) = -u$ or $1/u$) flips every ranking, so it describes opposite preferences.
Worked example: Which functions describe the same preferences?
$U(x, y) = xy$ for $x, y > 0$. Do these describe the same preferences? $V_1 = \ln x + \ln y$, $V_2 = 3xy + 5$, $V_3 = -xy$, $V_4 = x + y$.
- $V_1 = \ln(xy) = \ln U$, and $\ln$ is increasing: same preferences.
- $V_2 = 3U + 5$ is increasing in $U$: same preferences.
- $V_3 = -U$ is decreasing in $U$: it reverses every ranking.
- $V_4$ is not a function of $U$ at all. For example $(1, 9)$ and $(3, 3)$ have $V_4 = 10$ and $6$, but $U = 9$ for both: $V_4$ ranks a pair that $U$ calls a tie.
Answer: $V_1$ and $V_2$ do; $V_3$ and $V_4$ don't.
Worked example: Same curve, new label
Kara's preferences are described by $U = xy$ and by $V = 2(xy)^2 + 1$. Which level of V is the indifference curve $U = 5$?
- Write V in terms of U: $V = 2U^2 + 1$.
- On the curve $U = 5$: $V = 2(25) + 1 = 51$.
- Same bundles, same curve; only the label changed.
Answer: $V = 51$.
Common mistake
Treating utility as measured happiness. If $U(A) = 20$ and $U(B) = 10$, it is tempting to say A is "twice as good." But $V = \ln U$ describes the same person with values about 3.0 and 2.3, and $V = U^3$ gives 8,000 and 1,000. The ratio means nothing; only the order does. Check: a statement about preferences is meaningful only if every increasing transformation of $U$ leaves it true.
Ordinal utility is all we need for choices without risk. When we compare risky prospects in Expected Utility, the scale matters more: there a utility function is pinned down up to a positive linear change $aU + b$ with $a > 0$.