Question confusing me about Asset Market and Arrow-Debreu Equilibrium

[TonyAlmeidaAtLSE]

In the asset market equilibrium and Arrow-Debreu equilibrium, for each every agent, the budge constraint has the form of inequality (not strictly), however, Asset market clear is the aggregate asset is zero and market clear is aggregate endowment euqals the aggregate consumption. I wonder if we do Not assume the increasing utility function (local nonsatiation), is it possible that the budget constraint is Not binding while the condition of asset market clear and market clear satisfied respectively, say, some individual's budget constraint is Not binding while the aggregate consumption is equivalent to aggregate endowment?

 

[Ontoplankton]

I'd say not, because:

If for one individual the budget constraint isn't binding, then the total value of his demand is strictly less than the total value of his endowments. For everyone else, the total value of their demands is at most equal to the total value of their endowments (otherwise they would violate their budget constraints). So the total value of demands aggregated across all individuals is then strictly less than the total value of all endowments. This means the sum of all demands can't be equal to the sum of all endowments, because then their values would be the same.

(By "value", I mean sum of (price times amount of good).)

 

[R0man]

The concept of asset market equilibrium and Arrow-Debreu equilibrium can be confusing, especially when considering the budget constraint and market clearing conditions. In these equilibria, the budget constraint for each agent is typically represented as an inequality (not strictly), while the market clearing condition is that the aggregate assets must be equal to zero and the aggregate endowment must equal the aggregate consumption.

However, you raise a valid question about the possibility of the budget constraint not being binding while still satisfying the market clearing conditions. This scenario is possible, even without assuming increasing utility function or local nonsatiation. In this case, some individuals may have a budget constraint that is not binding, meaning that they have more resources than they need to consume their endowment. This excess can then be used to satisfy the market clearing conditions for the aggregate economy.

In essence, the budget constraint is not always a strict constraint in these equilibria. It is possible for some individuals to have a surplus of resources, which can then be used to balance out any deficits in the aggregate economy. This highlights the importance of considering the overall economy and not just individual agents when analyzing asset market and Arrow-Debreu equilibria.

In conclusion, while it may seem counterintuitive, it is possible for the budget constraint to not be binding while still satisfying the market clearing conditions in asset market and Arrow-Debreu equilibria. This further emphasizes the complexity of these equilibria and the need for a holistic approach when analyzing them.