Are Cobb Douglas goods complements or substitutes? Given 
$$U(x,y)= x^\alpha y^{1-\alpha}$$
$\alpha \in (0,1)$, 
are Cobb Douglas goods (here $x$ and $y$) complements, substitutes, or neither? Why? 

An explanation with mixed partial derivatives would be great. My calculus is better than my economics skills.
Try to think what you mean when you ask whether they're complements or substitutes. 

You could mean: "Does my marginal utility in $x$ increase when I get more $y$? That would correspond to the cross derivative $\frac{\partial U^2}{\partial x \partial y}$.

You could (and this is the convention) mean the response to a change in prices. Denote with stars the bundle that maximizes utility given a budget constraint.

$$x^* (p_x, p_y), y^*(p_x, p_y) = \arg\max_{x,y} U(x,y) + \lambda(Y - p_x x - p_y y) $$

Then, the goods are complements if you increase the price of one of the goods, and the demand of the other one decreases:

$$ \frac{\partial x^* (p_x, p_y)}{\partial p_y} &lt; 0 \\
\frac{\partial y^* (p_x, p_y)}{\partial p_x} &lt; 0 $$

And substitutes if that is larger than zero. For the case where that partial is exactly zero, the elasticity of substitution is zero. Try out yourself which case is correct for Cobb-Douglas!