<p>Consider an economy with two commodities. Production is trivial, <span class="math-container">$Y=\{0\}$</span>, there is a single consumer with endowment <span class="math-container">$(1,1)$</span> whose preferences are represented by the utility function given by <span class="math-container">$u(x_1,x_2)=\max\{x_1,x_2\}+1/2\cdot x_1 +1/2\cdot x_2$</span>. These preferences are continuous, strictly monotone, but not convex. You can verify that there is no equilibrium in this economy by showing that for no price system, the demand of the consumer equals the supply of <span class="math-container">$(1,1)$</span>.</p>

CC BY-SA 4.0
2020-12-07T13:35:40.333Z
Thank you. Your logic still hold when there is no endowment and only production makes supply?
martian03
CC BY-SA 4.0
2020-12-07T13:42:19.857Z
You usually need some input to produce some output.
Michael Greinecker
CC BY-SA 4.0
2020-12-07T13:13:32.077Z
2020-12-07T13:13:32.077Z
Michael Greinecker
<p>Consider a production economy with <span class="math-container">$L$</span> goods, a single consumer and a single producer whose production set are given by <span class="math-container">$Y\subset R^L$</span>. Question is to find the existence condition of equilibria of this economy.</p>
<p>I think the existence conditions are:</p>
<ol>
<li><p>continuity and monotonicity for consumer preference</p>
</li>
<li><p>convexity for firm's technology.</p>
</li>
</ol>
<p>Generally, convexity of consumer's preference seems to be required also for existence of equilibria. However, I cannot find the counter example that there is no equilibrium of this economy if consumer's preference is not convex and all other consumptions are satisfied.</p>
<p>Convexity of consumer's preference is really a condition of existence of equilibria in this one consumer-one producer economy?</p>

CC BY-SA 4.0
2020-12-07T11:48:02.160Z
Strict convexity is the condition of uniqueness of equilibrium. However, this is question for existence.
martian03
CC BY-SA 4.0
2020-12-07T12:37:12.047Z
I also know the counterexample that there is no equilibrium if firm's technology is not convex. However, this question is about convexity of consumer's preference, not firm.
martian03
<preferences><general-equilibrium><competitive-equilibrium><aggregate-production>
preference convexity and existence of equilbria
martian03
# preference convexity and existence of equilbria

(Asked by: username_0 on 2020-12-07)

Consider a production economy with $L$ goods, a single consumer and a single producer whose production set are given by $Y\subset R^L$. Question is to find the existence condition of equilibria of this economy.

I think the existence conditions are:

1. continuity and monotonicity for consumer preference
2. convexity for firm's technology.

Generally, convexity of consumer's preference seems to be required also for existence of equilibria. However, I cannot find the counter example that there is no equilibrium of this economy if consumer's preference is not convex and all other consumptions are satisfied.

Convexity of consumer's preference is really a condition of existence of equilibria in this one consumer-one producer economy?

**Tags:** preferences, general-equilibrium, competitive-equilibrium, aggregate-production

> **Comments:**
> **username_0**: Strict convexity is the condition of uniqueness of equilibrium. However, this is question for existence.  
> **username_0**: I also know the counterexample that there is no equilibrium if firm's technology is not convex. However, this question is about convexity of consumer's preference, not firm.  

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## 1 answers

### Answer by username_1 on 2020-12-07. Score: 4 | Accepted answer

Consider an economy with two commodities. Production is trivial, $Y=\{0\}$, there is a single consumer with endowment $(1,1)$ whose preferences are represented by the utility function given by $u(x\_1,x\_2)=\max\{x\_1,x\_2\}+1/2\cdot x\_1 +1/2\cdot x\_2$. These preferences are continuous, strictly monotone, but not convex. You can verify that there is no equilibrium in this economy by showing that for no price system, the demand of the consumer equals the supply of $(1,1)$.

> **Comments:**
> **username_0**: Thank you. Your logic still hold when there is no endowment and only production makes supply?  
> **username_1**: You usually need some input to produce some output.  

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