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I am just beginning to study some stuffs outside introductory/sophomore(?) micro/macroeconomics. And I met with a stuff called Arrow-Debreu model.

The question is,

1) What would be the proof that Arrow-Debreu model has many equilibra in general case? Can anyone also explain the criteria required for the model to have a unique equilibrium? (thorough explanation would be appreciated)

2) Can introductory model of basic supply and demand curve/model (so not Arrow-Debreu model, but the model used for economics beginners) have many equilibria? I do not think so, but unsure. If not, or if so, can anyone show me the proof?


3) Can anyone also thoroughly explain what Sonnenschein-Mantel-Debreu theorem is mathematically?

4) If a model has many equilibria, does this mean that the model has unstable solutions?

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  1. There exist explicit examples of economies in which one can calculate equilibria and that have multiple equilibria. It is also a straightforward consequence of the MSD-theorem. If all goods are gross substitutes, equilibrium is unique. This is extremely restrictive though. Some people derived uniquencess from assumptions on the distribution of characteristics of agents. Werner Hildenbrands work on the law of demand is the go-to stuff for this.

  2. No there cannot be multiple equilibria under the usual assumption that demand is strictly decreasing in price and supply strictly increasing. If the supply equals demand, we have $S(p)-D(p)=0$ and $S-D$ is a strictly increasing function which cannot have more than one zero.

  3. Just read it on Wikipedia. A consructive proof by Geanakoplos can be found here and a nice variant with a nice proof due to McFadden, Mas-Colell, Mantel, and Richter can be found here. A nice textbook treatment is in "Equilibrium Theory" by Hildenbrand and Kirman.

  4. It depends on what you consider to be "stability". The theory is essentially static.

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