I was wondering if anyone had any insight into the conditions that lead to a unique equilibrium in an exchange economy under a general equilibrium framework. More specifically, I know that the two "general" conditions that can be used to show an equilibrium is unique are Gross Substitutes and Weak Axiom of Revealed Preference.
But beyond that, I'm having trouble figuring out what those two conditions actually look like in practice, and finding examples for each case. For example, a few of the specific questions I have (and note, parts of my questions may themselves be wrong, so please do not hesitate to point out any and all flaws you see!):
In an Edgeworth Economy setting, both consumers having Cobb Douglas utility functions is sufficient, correct? I'm not 100% sure, but I think that satisfies the gross substitutes condition. That said, is that result generalize-able? If all individuals have Cobb Douglas utilities in an n-consumer, l-good case, is there still a unique equilibrium?
Similarly, in an Edgeworth Economy setting, if both individuals have quasilinear utilities in the same good it is also sufficient to show a unique equilibrium (if it exists). Is that result generalizable to an n case?
Moreover, if anyone else has any good examples of either utility function forms or tips regarding proving or disproving uniqueness in a given case, I'd love to hear them! I'm particularly curious about Edgeworth Economy cases, but any general examples/results would be wonderful!
Thanks so much!