If $X$ is a Giffen good then $Y$ must be a normal good While going through some problems as part of self-study I encountered the following True/False question:  

Q. Steven only consumes two goods: $X$ and $Y$. If $X$ is a Giffen good for Steven, then $Y$ must be a normal good for Steven.

The given answer is True.

I am unable to understand why it is necessary for the other good to be normal. Why can't it be inferior/Giffen?
Reason: Both goods cannot be inferior.

Let's say originally you consume $x$ and $y$. So your budget constraint looks like

$$p_x x + p_y y = I.$$

If both X and Y are inferior, when income goes down from $I_0$ to $I'$, the quantity demanded for both has to go up (by definition) from $x$ to $x'$ and $y$ to $y'$. This implies

$$p_x x' + p_y y' = I' &lt; I = p_x x + p_y y.$$

This is a contradiction, since $x' &gt; x$ and $y' &gt; y$.

Your question is a bit more specific. But since a Giffen good must be inferior, this answers your question as well.