# Intuition for why $EV>CV$ for a normal good

I understand the mathematical proof and the graphical illustration behind this property ($EV>CV$ for the variation in the price of a normal good), but I still do not understand the economic intuition. Can anyone explain, in intuitive terms, why the comparison between $EV$ and $CV$ is given by the sign of the wealth effects?

Edit: there seems to be different definitions of $EV$ and $CV$. Here are the ones that I use. Following a change in price from $p^{0}$ to $p^{1}$, the income being fixed at $w$, $EV$ is defined by \begin{equation*} v(p^{0},w+EV) = v(p^{1},w) \end{equation*} whereas $CV$ is defined by \begin{equation*} v(p^{0},w) = v(p^{1},w-CV). \end{equation*} For these definitions, if I am not mistaken, if the good is normal, then $EV>CV>0$ for a price decrease and $0>EV>CV$ for a price increase.

For a normal good, for an increase in the price of good $x$, the compensating variation must be greater than or equal to the equivalent variation because once $p_x$ has increased, with $p_y$ remaining constant at $1$, it must cost more to compensate the consumer in get them back to their original indifference curve than would be required to take from them at the original price level to take them from the old to the new indifference curve, because both goods are at least or more expensive than before (another way of saying this is that the marginal utility that the consumer gets from each additional unit of cash income must be lower at the new price level, so more cash income must be given to get the consumer back to their original utility level). For a decrease in the price of a normal good, this result is reversed.
• Thanks for the answer. We probably work with different definitions: according to the one I work with, for a normal good, $EV>CV>0$ for a decrease in the price, and $0>EV>CV$ for an increase in the price. I still have trouble understanding the intuition in spite of your explanation, so I'll wait for other answers. – Oliv Jan 2 '18 at 16:09