<p>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 $\omega_n &gt; 0$ for all $n$ (where $\omega_n$ is the total endowment of good $n$).</p>

<p>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?</p>

CC BY-SA 3.0
2016-02-20T11:05:55.817Z
Try a fixed point theorem e.g. brouwer's
BB King
CC BY-SA 3.0
2016-02-20T11:15:27.423Z
What do you mean you are trying to demonstrate the conditions? Are you trying to show that $U^i$ is strictly concave?
Giskard
CC BY-SA 3.0
2016-02-21T09:57:44.150Z
@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
PostDocing
CC BY-SA 3.0
2016-06-06T03:03:16.433Z
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
<microeconomics><mathematical-economics>
Existence of a Competitive Equilibrium
PostDocing
# Existence of a Competitive Equilibrium

(Asked by: username_0 on 2016-02-20)

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?

**Tags:** microeconomics, mathematical-economics
> **Comments:**
> **username_1**: Try a fixed point theorem e.g. brouwer's  
> **username_2**: What do you mean you are trying to demonstrate the conditions? Are you trying to show that $U^i$ is strictly concave?  
> **username_0**: @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  
> **username_3**: These are sufficient conditions for existence,not necessary conditions. Either way, the proof of sufficiency is not trivial. Look at graduate micro textbooks.  

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