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Recall that the slutsky equation is:

$$\frac{\partial x_i}{\partial p_i}=\frac{\partial h_i}{\partial p_i}-\frac{\partial x_i}{\partial m}x_i$$

I know $\frac{\partial h_i}{\partial p_i}$ defined as the substitution effect, the second part of the equation is the income effect.

My Question: is the income effect defined as $-\frac{\partial x_i}{\partial m}x_i$ or $\frac{\partial x_i}{\partial m}x_i$ (is the negative sign a defining characteristic?)

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The income effect is defined as $\mathbf{-\frac{\partial x_i}{\partial m}x_i}$.

Let $x_i$ be a normal good; that is, a good whose Marshallian demand increases with an increase in income $\left(\frac{\partial x_i}{\partial m} > 0\right)$. Even if there was no substitution effect $\left(\frac{\partial h_i}{\partial p_i} = 0\right)$ from an increase in own-price, the amount of $x_i$ consumed would still be reduced as the consumer is effectively poorer. Hence the net income effect in this case is negative, which necessitates the negative sign since $x_i$ is non-negative. Analogous logic holds if $x_i$ is an inferior good instead.

This interpretation is consistent with Nicholson & Snyder (p. 156).

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