Are Cobb-Douglas preferences monotone according to the marginal utility condition? I understand that Cobb-Douglas preferences represented by $U(x,y)=x^ay^b$ are strictly monotonic, because increasing at least one of the goods in the bundle increases utility.
However, another definition of strict monotonicity says that marginal utility of each good should be strictly positive. This is not the case with the above function at $(0,0)$. Is there a way to resolve this?
They do not satisfy either condition. An increase in the quantity of one good need not increase utility; $U(0,1)=0^a1^b=0=0^a2^b=U(0,2)$.
The condition that all marginal utilities must be positive is inherently problematic because it does depend not just on the underlying preferences. For one, not every utility representation needs to be differentiable. But even that is not enough. Suppose there is a single good and more is better than less. You can represent these preferences by the utility function given by $U(x)=x$. Now $U'(x)=1$ for all $x$, so these preferences are supposedly strictly monotone. But now consider the utility function given by $V(x)=(x-1)^3$. If $x'&gt;x$, then $V(x')&gt;V(x)$, more is better and $V$ represents the same preferences as $U$. But $V'(1)=0$, so the preferences are not strictly monotone according to the definition in terms of marginal utilities.
The problem just mentioned is actually even worse. Let $U$ be a differentiable utility function on $\mathbb{R}_+^2$ with strictly positive partial derivatives everywhere. Let $(x^*,y^*)\gg0$be any commodity bundle. Define $V$ by $$V(x,y)=\big(U(x,y)-U(x^*,y^*)\big)^3.$$ $V$ represents the same preferences as $U$ but the partial derivatives at $(x^*,y^*)$ are zero.
Long story short: Every partial derivative being strictly positive is a sufficient but not necessary condition for a differentiable function to be increasing in every coordinate.
Yet another issue is that it is not entirely clear how one defines the derivative at the boundary of the commodity space. There are different notions of differentiability which need not be equivalent.