We can't see indifference curves, but we can see choices. If a consumer picks the best bundle in her Budget Sets, as in The Best Affordable Bundle, then every choice tells us something: the chosen bundle beat everything else she could afford.
Key idea
If bundle A is chosen when bundle B (different from A) was affordable, A is revealed preferred to B. The weak axiom of revealed preference: if A is revealed preferred to B, then B is never chosen when A is affordable.
Why it works
Suppose she chose A at prices $p$ while B cost no more than A. If her preferences were consistent and she picked her favorite affordable bundle, then $A \succeq B$, and with a strictly bowed indifference map, $A \succ B$. If at other prices she later chose B while A was affordable, the same reasoning gives $B \succ A$. Both can't hold, so a consumer with well-behaved preferences can't produce that pair of choices.
Worked example: Consistent choices
At prices (2, 4) Sam chose A = (10, 5), spending \$40. At prices (4, 2) he chose B = (4, 12), spending \$40. Check the weak axiom.
- At the first prices, B costs $2(4) + 4(12) = 56 > 40$: not affordable, so the first choice says nothing about B.
- At the second prices, A costs $4(10) + 2(5) = 50 > 40$: not affordable either.
- Neither is revealed preferred to the other: no contradiction.
Answer: Consistent.
Worked example: A violation
At prices (2, 4) Tara chose A = (10, 5), spending \$40. At prices (4, 2) she chose B = (14, 2), spending \$60. Check the weak axiom.
- At the first prices, B costs $2(14) + 4(2) = 36 \le 40$: B was affordable, so A is revealed preferred to B.
- At the second prices, A costs $4(10) + 2(5) = 50 \le 60$: A was affordable, so B is revealed preferred to A.
- Each is revealed preferred to the other, and A and B differ. No consistent preferences that she maximizes could produce both choices.
Answer: A violation of the weak axiom.
Common mistake
Checking only whether the two chosen bundles differ, or whether one has more of each good. The test is about affordability: price each chosen bundle at the other situation's prices and compare with what was spent there. Check: write the two costs side by side, "cost of B at prices 1 versus spending 1" and "cost of A at prices 2 versus spending 2." A violation needs both to be affordable.
Revealed preference was introduced by Paul Samuelson in 1938. With many observations, the generalized axiom (Afriat, 1967) tests whether any well-behaved utility function could have produced a whole data set of choices, and it is used to test real spending data.