# 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?

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:

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

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