What is the relation of WARP to utility functions?

If x(p,y) is generated by utility maximization, then it satisfy WARP:

1. Let x⁰ maximize utility at p⁰ and x¹ at p¹.

2. Suppose p⁰x¹≤p⁰x⁰

3. Since x¹ is not optimal at p⁰, u(x¹)<u(x⁰)

4. Then x⁰ cannot be affordable at p¹: p¹x⁰>p¹x¹.

- If there are two goods, then WARP and budget balancedness imply that there is a utility function that rationalizes the choices.

- for more than two goods: SARP is needed.

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