<p>I have been trying to figure out conditions on the preferences of traders in an exchange economy such that the equilibrium prices are unique but I cannot be sure since all sources seem to be contradicting each other. On the <a href="https://en.wikipedia.org/wiki/Arrow%E2%80%93Debreu_model" rel="nofollow noreferrer">Arrow-Debreu wikipedia page</a> it says &quot;In general, there may be many equilibria; however, with extra assumptions on consumer preferences, namely that their utility functions be strongly concave and twice continuously differentiable, a unique equilibrium exists.&quot; However, the Sonnenschein–Mantel–Debreu theorem does not tells us that the excess demand curve inherits the shape of the utility functions.</p>
<p>I do understand that for utilities satisfying the gross substitutes conditions the equilibrium prices are unique but are prices unique if we assume concave and twice differentiable utility functions?</p>

CC BY-SA 4.0
2020-07-29T22:57:31.853Z
Does this answer your question? [Pure exchange economy: Given an initial endowment are multiple equilibria possible?](https://economics.stackexchange.com/questions/19289/pure-exchange-economy-given-an-initial-endowment-are-multiple-equilibria-possib)
Giskard
CC BY-SA 4.0
2020-07-30T07:17:08.183Z
What is written on Wikipedia about uniqueness is simply wrong. The most common sufficient condition for equilibrium uniqueness is that all goods are gross substitutes for every consumer.
Michael Greinecker
<general-equilibrium>
Uniqueness of Arrow-Debreu Equilibria
Denizalp
# Uniqueness of Arrow-Debreu Equilibria

(Asked by: username_0 on 2020-07-29)

I have been trying to figure out conditions on the preferences of traders in an exchange economy such that the equilibrium prices are unique but I cannot be sure since all sources seem to be contradicting each other. On the [Arrow-Debreu wikipedia page](https://en.wikipedia.org/wiki/Arrow%E2%80%93Debreu_model) it says "In general, there may be many equilibria; however, with extra assumptions on consumer preferences, namely that their utility functions be strongly concave and twice continuously differentiable, a unique equilibrium exists." However, the Sonnenschein–Mantel–Debreu theorem does not tells us that the excess demand curve inherits the shape of the utility functions.

I do understand that for utilities satisfying the gross substitutes conditions the equilibrium prices are unique but are prices unique if we assume concave and twice differentiable utility functions?

**Tags:** general-equilibrium

> **Comments:**
> **username_1**: Does this answer your question? [Pure exchange economy: Given an initial endowment are multiple equilibria possible?](https://economics.stackexchange.com/questions/19289/pure-exchange-economy-given-an-initial-endowment-are-multiple-equilibria-possib)  
> **username_2**: What is written on Wikipedia about uniqueness is simply wrong. The most common sufficient condition for equilibrium uniqueness is that all goods are gross substitutes for every consumer.  

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