# Is local non-satiation enough to talk about Walrasian Equilibrium being subset of Core

In Advanced Microeconomic Theory, the author mentions that the Cobb Douglas utility function is neither strongly increasing nor strictly quasiconcave over all of R+. But the condition of strongly increasing in R+ is used to prove that set of Walrasian Equilibrium allocations is a subset of Core. So we can't apply this result to Cobb Douglas utility functions to prove that Walrasian Equilibrium obtained is part of core. Is local non satiation sufficient to say set of Walrasian Equilibrium allocations is a subset of Core as Cobb Douglas satisfies local non satiation?

• Depending on how you define the core, no assumption whatsoever is needed. You don't need any assumption to show that not every consumer can be made strictly better off in a Walrasian equilibrium. – Michael Greinecker Apr 15 '20 at 19:56

As a side note: Cobb-Douglas preferences do satisfy strict monotonicity on $$\mathbb{R}^{n}_{++}$$. Thus, if you can argue that you may restrict attention to bundles where all entries are strictly positive, the arguments based on strict monotonicity applies.