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The Walrasian demand function $x(p,w)$ satisfies the weak axiom of revealed preference if the following property holds for any two price wealth situations $(p,w)$ and $(p',w')$: If $p•x(p',w')\le w$ and $x(p',w')\ne x(p,w)$, then $p'•x(p,w)>w'$. Where $x$ is the amount of commodity, $p$ is price and $w$ is wealth.

The book(MWG's Microeconomics Theory, Page 29-30) says that the graph below satisfies the weak axiom. I have difficulty in understanding the graph. To be specific, I don't know why there is a dot in each of the budget set and I can't understand why it satisfies the weak axiom. enter image description here

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So WARP says that if bundle x' you choose under budget B' is affordable at budget B'' (and you don't choose x' - i.e x'$\not=$x''), then x'' cannot be affordable under budget B'.

The picture you showed gives the 2 bundles that the person chooses. Those are given in the picture. As you can see, x' is not affordable at budget B'' and bundle x'' is not affordable at budget B', so WARP is satisfied.

If you are reading MWG and you don't understand what those dots represent in the consumption space, then I would recommend reading more from the beginning of chapter 2.

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