# Existence of a Competitive Equilibrium

My professor gives us these conditions for the existence of a Competitive Equilibrium (CE) in a pure exchange economy: $U^i$ is continuous, strictly increasing and strictly concave for all $i$ and $\omega_n > 0$ for all $n$ (where $\omega_n$ is the total endowment of good $n$).

I am trying to demonstrate these conditions but I am a bit stuck. I started using the Maximum theorem but now I cannot go ahead. Any hint?

• Try a fixed point theorem e.g. brouwer's – BB King Feb 20 '16 at 11:05
• What do you mean you are trying to demonstrate the conditions? Are you trying to show that $U^i$ is strictly concave? – Giskard Feb 20 '16 at 11:15
• @denesp I am trying to show that these are the conditions for the existence of a competitive equilibrium. Thus, that $U^i$ must be strictly concave, increasing and continuous – PhDing Feb 21 '16 at 9:57
• These are sufficient conditions for existence,not necessary conditions. Either way, the proof of sufficiency is not trivial. Look at graduate micro textbooks. – Michael Greinecker Jun 6 '16 at 3:03