WARP (Demand Function)

Definition

For UMP x(p, w):
x(p, w) satisfies the weak axiom of revealed preference (WARP, or WA) if for any two (p, w) and (p', w'):
If
and ,
then

Interpretation
If I choose to buy x(p, w) when x(p', w') is available, then x(p, w) cannot be affordable when x(p', w') is chosen.

Or make it more clear:
Given (p, w), if I choose to buy x(p, w), not x(p', w'): , recall in WARP for choice structure, when x, y in A and x in C(A). A is the budget set B(p, w)
When x(p', w') is also affordable under (p, w): , recall in WARP for choice structure, it means x, y in A.
Then x(p, w) is not affordable given (p', w'): , recall in WARP for choice structure, if x, y in B, we cannot have y in C(B) but not x in C(B). if we have y in C(B) but not x in C(B), x is not in B. B is the budget set B(p', w')

Proposition

Suppose that x(p, w) is homogeneous of degree zero and satisfies Walras's law. Then x(p, w) satisfies the weak axiom if and only if the following property holds:

For any compensated price change from an initial situation (p, w) to a new price wealth pair we have

with strict inequality whenever x(p, w) ≠ x(p', w')

Note

Example