In a two-good world, if one good is a Giffen good then the other is a good luxury

Question

In a two-good world, if one good is a Giffen good then the other is a good luxury.

This is a false statement. Reason: If X is a Giffen good then Y must be a normal good. But we cannot get any conclusion about luxury good. Am I correct?

Answer 1

If $X$ is a Giffen good, then it must an inferior good.

Next, as the share-weighted sum of the income elasticities of all goods sum to 1 (Engel aggregation), and given that the income elasticity of $X$ is negative, the income elasticity of good $Y$ must be bigger than 1, which means that $Y$ is a luxury good.

To see that the latter, start from the budget constraint: $$ p_X X(p,m) + p_Y Y(p,m) = m. $$ Take the derivative with respect to income, $m$: $$ p_X \frac{\partial X}{\partial m} + p_Y \frac{\partial Y}{\partial m} = 1. $$ This gives: $$ \frac{p_X X}{m} \frac{\partial X}{\partial m} \frac{m}{X} + \frac{p_Y Y}{m} \frac{\partial Y}{\partial m}\frac{m}{Y} = 1. $$ Let $s_X = \frac{p_X X}{m}$ be the budget share of good $X$ and $s_Y = \frac{p_Y Y}{m}$ be the budget share of good $Y$ and denote by $\varepsilon^X_m$ and $\varepsilon^Y_m$ the income elasticities of goods $X$ and $Y$ respectively. Then: $$ s_X \,\,\varepsilon^X_m + s_Y \,\,\varepsilon^Y_m = 1. $$ The first term is negative (as $X$ is inferior, $\varepsilon^X_m$ is negative). This means that $s_Y \varepsilon^Y_m \ge 1$. As $s_Y \le 1$, we obtain: $$ \varepsilon^Y_m \ge \frac{1}{s_Y} \ge 1. $$ This shows that $Y$ is a luxury good.