# Example in which revealed preference doesn't imply preference

I'm studying Varian's Intermediate Microeconomics, in which the principle of revealed principle is stated as follows (for now we're assuming strictly convex indifference curves):

Let $(x_1, x_2)$ be the chosen bundle when prices are $(p_1, p_2)$, and let $(y_1, y_2)$ be some other bundle such that $p_1x_1+p_2x_2 \geq p_1y_1+p_2y_2$. Then if the consumer is choosing the most preferred bundle she can afford, we must have $(x_1,x_2) \succ (y_1,y_2)$.

It's further stated that "if the consumer chooses the best bundles she can afford, then the revealed preference implies preference". I can't get an intuitive grasp on this. Even if a bundle $X$ is chosen to $Y$ $(Y \neq X)$, and even if $X$ isn't the best affordable bundle, shouldn't that at least imply that $X \succ Y$? Because if not, then there must be some example where $Y \succeq X$ and still the consumer chose $X$.

Is there some counterexample in which revealed preference of $X$ does not imply preference, provided that the chosen bundle $X$ isn't the best affordable one?

• That makes sense, but I was confused because earlier in the "Revealed Preferences" section, the author says, "Suppose that we are willing to postulate that this consumer is an optimizing consumer...". So then if the consumer is perfectly rational, is it safe to assume that revealed preference of $X$ to $Y$ will always imply preference of $X$ to $Y$? – user9343456 Mar 25 '17 at 12:49