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?

Question

Since the concavity of the Engel Curve determines whether it is a necessity or luxury (i.e. how fast quantity demand changes in relation to changes in income), and since the second derivative of a Cobb-Douglas Engel Curve is 0, does that mean it is neither category?

Edit: In simple terms, if an Engel Curve is a straight positively-sloped line, it is obviously a normal good. But if the curve is represented by a function like so:

$I = 10 * P_x * x.$

Then the curve is ambiguously sloped. If the price of $x$ ($P_x$) happens to be greater than 1/10 then the good is a luxury, and vice versa for a good with $P_x < 1/10$. But if it's ambiguous like this, then the second derivative (which indicates concavity and therefore what direction the curve is increasing/decreasing in) then is it impossible to tell?

For reference, this Engel Curve was derived from the Cobb-Douglas utility function:

$U(x,y) = x^{(1/10)}y^{(9/10)}.$

Answer 1

Recall the following equivalent definitions for luxury goods and necessities:

• A good $x$ is considered a necessity if $e_{(x,I)}<1$.

• A good $x$ is considered a luxury good if $e_{(x,I)}>1$.

  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.