# Second welfare theorem: can it be used to show there does not exist any competitive equilibrium? (exchange economies)

The one version of the Second Welfare Theorem states that: if there exists a competitive/Walrasian equilibrium and an endowment $$X$$ is Pareto efficient, then there is a price vector $$\hat{P}$$ for which $$(\hat{P}, X)$$ is a competitive equilibrium.

If I have a single allocation $$\bar{X}$$ that is Pareto efficient but not a competitive equilibrium (ie there is no $$\hat{p}$$ that makes this allocation maximize all players' utility), does this imply there is no competitive equilibrium?

What if all feasible $$X$$ are Pareto efficient? Would this prove that a competitive equilibrium does not exist?

Thanks very much

• Are you talking about exchange economies? And what would be a case in which every allocation is Pareto efficient? Jan 4, 2023 at 19:06
• Yes, exchange economies - I'll make that clearer now. The indifference curves coincide over the whole Edgeworth-box. The situation is the same as in this recent question - economics.stackexchange.com/questions/53869/… Jan 4, 2023 at 19:32

## 1 Answer

In every competitive equilibrium, a consumer's endowment is in their budget set. Since they choose an optimal point in the budget set, every consumer in a competitive equilibrium will receive a bundle that is at least as good as the endowment. If the endowment allocation is Pareto efficient, no consumer can receive a bundle that is strictly better than their endowment either. So everyone choosing their endowment from the budget set would be an optimal choice for them, and the endowment allocation is, of course, feasible. So if a competitive equilibrium exists from a Pareto efficient endowment allocation, the endowment allocation must be a competitive equilibrium allocation.

Consequently, if a Pareto efficient endowment allocation is not an equilibrium allocation, there cannot exist any equilibrium.

Consider a simple Edgeworth-box economy in which both consumers have preferences given by $$u(x_1,x_2)=x_1+x_2$$. Then both consumers consuming their endowment and prices satisfying $$p_1=p_2>0$$ is a competitive equilibrium. Here, every allocation (excluding disposal) is efficient.

• Great - thats very clear. Thank you very much. Jan 4, 2023 at 21:47