# Assumptions for the existence of a Walrasian equilibrium

I have a problem set stating that a competitive equilibrium does exist under a series of assumptions on the economy.

The question is "Show that the following six assumptions are needed for existence by constructing counterexamples to it where 5 of them are satisfied and one not".

The assumptions are continuity, convexity and local nonsatiation of preferences and closedness, convexity and inaction of the production set.

From what I got from the question and the TA explanations, it seems that if one of this properties fails, then there should be no competitive equilibrium. But I have actually found cases in which these properties are violated but an equilibrium exists. For example, in an Edgeworth box with bliss point preferences (which are not locally nonsatiated) we can find an equilibirum; the same happens when we have for example lexicographic (which are not continuous) and perfect substitutes.

SO my question is are these assumptions necessary for the existence of an equilibirum or not? And if not, what would be the purpose of the exercise?

• The point is that, if any one of those conditions are not satisfied, there may be examples without such an equilibrium, and your task is to find an example for each case. It is not suggesting that the failure of a condition guarantees that there will not be such an equilibrium – Henry Nov 10 '18 at 1:13
• @Henry So you would say these conditions are sufficient but not necessary for the existence of a Walrasian equilibrium? – PhDing Nov 10 '18 at 15:10
• That is what the question suggests, or more precisely both that they are sufficient and that no proper subset of them is sufficient – Henry Nov 10 '18 at 17:45
• @Henry Thanks! The TA corrected himself today and confirmed your answer. Thanks again! – PhDing Nov 10 '18 at 19:27