# Income effect $-\frac{\partial x_i}{\partial m} x_i$ or $\frac{\partial x_i}{\partial m}x_i$?

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?)

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.