Once a person can rank bundles, we can draw the ranking. Fix one bundle and collect every bundle she likes exactly as much. That set is an indifference curve. If her preferences come from a utility function $U(x, y)$, the curve is the graph of the equation $U(x, y) = k$, just as in A Graph Shows Every Solution: every point on it satisfies the equation, and no other point does.
Key idea
An indifference curve is a set of bundles with equal utility, $U(x, y) = k$. Under the usual assumptions the curves slope down, never cross, fill the whole graph (one through every bundle), and curves farther from the origin are preferred.
Why it works
Down-sloping: start at a bundle and add some $x$. Under more-is-better that is an improvement, so to get back to the same level you must take some $y$ away. Never crossing: suppose two curves met at bundle C. Pick A on the higher curve and B on the lower one. Then $A \sim C$ and $C \sim B$, so by transitivity $A \sim B$, even though A sits on the higher curve and is strictly preferred. That contradiction means curves of one person cannot cross. One through every bundle: completeness lets every bundle be compared with every other, so each one has its own set of equally good bundles.
Worked example: Finding a point on a curve
With $U(x, y) = xy$, bundle (4, 9) is on one of Grace's indifference curves. Which bundle with $x = 6$ is on the same curve?
- The curve through (4, 9) has $U = 4 \cdot 9 = 36$.
- Every bundle on it satisfies $xy = 36$. At $x = 6$: $6y = 36$, so $y = 6$.
- Check: $U(6, 6) = 36$. Grace gave up 3 units of $y$ for 2 more units of $x$ and is exactly as well off.
Answer: $(6, 6)$.
Worked example: A higher curve without more of everything
With $U(x, y) = xy$, compare $A = (2, 10)$ and $B = (5, 5)$.
- $U(A) = 20$ and $U(B) = 25$.
- B is on a higher indifference curve, so B is preferred, even though it has less $y$ than A.
- "Higher" means preferred, not "more of every good." The balanced bundle wins here because this person values variety (her curves bow toward the origin).
Answer: B is on the higher curve.
Common mistake
Drawing indifference curves that cross, or reading a higher curve as "more of every good." Curves of one person never cross (the transitivity argument above). And a bundle on a higher curve can contain less of one good, as B did. Check: to compare two bundles, compute their utilities (or find their curves). Only when one bundle has more of both goods can you rank them without the curves.
Indifference curves are the consumer's side of the picture. In Budget Sets we add what she can afford, and in The Best Affordable Bundle we put the two together.