<p>The following may work.</p>
<p>Let there be two consumers. Consumer 1 has standard (convex, continuous, strictly increasing) and let consumer 2 have constant utility (i.e. indifferent between any two consumption bundles). This still satisfies your condition.</p>
<p>Then given some endowments, and arbitrary prices, let consumer 1 maximize her utility. Then consumer 2 will (by definition) also maximize his utility as he is indifferent between anything. This will then give a competitive equilibrium.</p>
<p>The only Pareto efficient equilibrium, however, is when consumer 1 has all the endowments, which will usually not be the case.</p>

CC BY-SA 4.0
2021-09-20T16:07:31.353Z
I can't understand what exactly "consumer 2 has constant utility" means. What I thought is goods 2 is neutral to consumer 2 so his indifferent curve is vertical. Is that the same what you mean?
guest
CC BY-SA 4.0
2021-09-20T16:12:19.017Z
@guest No I meant that consumer 2 is indifferent between any two bundles. This means that he has constant utility function, e.g. $u(x) = 0$ for all consumption bundles $x$. His indifference curve is the entire consumption space.
tdm
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2021-09-20T16:14:15.393Z
I got it, and one more question. A vertical indifference curve also satisfies above condition too?
guest
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2021-09-20T16:19:06.247Z
yes. As long as indifference curves are not upward sloping, or inverted (i.e. they look normal but less is better), you should be good.
tdm
CC BY-SA 4.0
2021-09-20T15:54:45.347Z
2021-09-20T15:54:45.347Z
tdm
<p>I want to find an example that utility function is not increasing<br />
but still satisfies that <span class="math-container">$u(x) \geq u(y)$</span> for all <span class="math-container">$x \geq y$</span><br />
and 1st welfare theorem fails.<br />
I want to prove this on edgeworth box not using mathematical process.</p>
<p>Non-increasing(non-monotonic) utility function is related to non-satiation properties?</p>

<utility><general-equilibrium>
Failure of 1st welfare theorem with non-increasing utility function
guest
# Failure of 1st welfare theorem with non-increasing utility function

(Asked by: username_0 on 2021-09-20)

I want to find an example that utility function is not increasing  
but still satisfies that $u(x) \geq u(y)$ for all $x \geq y$  
and 1st welfare theorem fails.  
I want to prove this on edgeworth box not using mathematical process.

Non-increasing(non-monotonic) utility function is related to non-satiation properties?

**Tags:** utility, general-equilibrium

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## 1 answers

### Answer by username_1 on 2021-09-20. Score: 1 | Accepted answer

The following may work.

Let there be two consumers. Consumer 1 has standard (convex, continuous, strictly increasing) and let consumer 2 have constant utility (i.e. indifferent between any two consumption bundles). This still satisfies your condition.

Then given some endowments, and arbitrary prices, let consumer 1 maximize her utility. Then consumer 2 will (by definition) also maximize his utility as he is indifferent between anything. This will then give a competitive equilibrium.

The only Pareto efficient equilibrium, however, is when consumer 1 has all the endowments, which will usually not be the case.

> **Comments:**
> **username_0**: I can't understand what exactly "consumer 2 has constant utility" means. What I thought is goods 2 is neutral to consumer 2 so his indifferent curve is vertical. Is that the same what you mean?  
> **username_1**: @guest No I meant that consumer 2 is indifferent between any two bundles. This means that he has constant utility function, e.g. $u(x) = 0$ for all consumption bundles $x$. His indifference curve is the entire consumption space.  
> **username_0**: I got it, and one more question. A vertical indifference curve also satisfies above condition too?  
> **username_1**: yes. As long as indifference curves are not upward sloping, or inverted (i.e. they look normal but less is better), you should be good.  

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