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"For complete markets, the security market equilibrium is the same as the contingent commodity equilibrium but with far fewer spot markets."

I wrote this down in my notes today on what my professor said but I don't really understand why this is the case. This may be the wrong way of posing a question but I thought this was rather interesting. Any suggestions on the matter are greatly appreciated.

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You seem to be looking for information on the Radner equilibrium.

Radner equilibrium is an economic concept defined by economist Roy Radner in the context of general equilibrium. The concept is an extension of the Arrow–Debreu equilibrium and the base for the first consistent incomplete markets framework.

Further reading:
Jstor: Existence of Equilibrium of Plans, Prices, and Price Expectations in a Sequence of Markets
Handbook of Mathematical Economics: Chapter 20

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