If $X$ is a Giffen good then $Y$ must be a normal good

Question

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?

Answer 1

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

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

This is a contradiction, since $x' > x$ and $y' > y$.

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

Answer 2

If the other good is inferior, the indifference curve would be shift to the left and budget constraint shift to the right when there is an increase in income. That would make the consumer have no utility maximization point. cmiiw