If the Engel Curve of a Cobb-Douglas utility function is positive and linear, than does that mean it is neither a necessity nor a luxury good?
Recall the following equivalent definitions for luxury goods and necessities:

A good $x$ is considered a necessity if $e_{(x,I)}&lt;1$.

A good $x$ is considered a luxury good if $e_{(x,I)}&gt;1$.
As you can see, these definition do not encompass all possible scenarios, so any specific good does not have to be either a luxury or a necessity.


In the case of a Cobb-Douglas utility function $U(x,y)=x^\alpha \cdot y^\beta$ we get $x^* = \frac {\alpha I}{P_x}$. One can easily verify that $e_{(x,I)}=1$. In other words, the demand does for $x$ does not change with $I$. This means that in this case, $x$ is neither a luxury good nor a necessity.