Revealed Preference

Letting choices speak

Usually we start with preferences and derive demand. Revealed preference runs it backwards: we observe what someone buys at different prices and incomes, and ask what that tells us about their preferences — without ever writing down a utility function.

The key idea. If bundle A was chosen when bundle B was also affordable, the consumer directly revealed A preferred to B — they could have had B and picked A instead. A rational consumer can’t contradict themselves, which gives one testable rule:

Weak Axiom of Revealed Preference (WARP). If A is revealed preferred to B, then B must never be revealed preferred to A.

A violation is catching the consumer both ways at once:

  • in situation 1 they pick A while B was affordable, and
  • in situation 2 they pick B while A was affordable.

No single set of preferences could rank A over B and B over A — so the behavior is inconsistent with utility maximization.

When can that even happen? Only if the two budget lines cross, so that each chosen bundle can lie inside the other situation’s budget. Here the two budgets are fixed and crossing:

  • Situation 1 (blue): good 1 costs $2, good 2 costs $1, income $20.
  • Situation 2 (green): good 1 costs $1, good 2 costs $2, income $20.

Try it. Each slider sets the units of good 1 in the chosen bundle (good 2 is whatever the budget allows). The red stretch of each line is the “danger zone” — a bundle there was affordable in the other situation. Put both chosen dots on their red zones and WARP breaks. The Show a violation button does it in one click.

What to notice

  • The red zones are the whole story. A bundle on the red part of a line was affordable in the other situation. A WARP violation needs both dots on their red zones at once.
  • Click Show a violation: A sits at (8, 4) and B at (4, 8). Each was affordable when the other was chosen — a flat contradiction. Click Show a consistent choice: B slides out to lots of good 1, off its red zone, and the contradiction disappears.
  • This is why the two budgets must cross. If one budget lay entirely inside the other, every choice would be explainable — you could never catch the consumer both ways, and WARP could never be violated.