# 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.

I think the magnitude of EV is less than the magnitude of CV for a normal good which increases in price (and the reverse for a normal good which reduces in price, so there may be a sign issue)

Intuitively, you should be able to argue that for a small price increase for an income-inelastic good (so neither normal nor inferior) the increase in income based on the new price you would need to take you back to your old utility (CV: the Compensating Variation) should be equal to the amount based on the old price you would be prepared to pay to avoid the price change (EV: the Equivalent Variation). The shift along the utility curves caused by the price change is similar in both cases as the unchanged-priced goods substitute for the increased-price good. Graphically the two utility curves are in a sense almost equally spaced with a vertical translation

Meanwhile with a normal good, you now face a increased price for this good, so in the CV calculation not only do you need an increased income to compensate for the loss of utility, but it has to be even higher because you will want more of this increased-price normal good when you get your compensation than you would if it were income-inelastic, so CV is even higher for a normal good than for an income-inelastic good for a given level of EV calculated on the old prices. For a decrease in prices you would get the opposite effect

Some Oxford lecture notes (pages 5 and 6) use a similar argument:

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
• @Oliv Your inequalities are consistent with my "the magnitude of EV is less than the magnitude of CV for a normal good which increases in price (and the reverse for a normal good which reduces in price" – Henry Jan 3 '18 at 2:28