# General Equilibrium Involving Production

I need a little conceptual clarification. For a standard $$N*K*M$$ general equilibrium model, would an allocation, say, $$y^k$$ be Pareto Optimal if it does not solve $$max(py^k)$$? I understand that the competitive allocations would Pareto Optimal given strictly convex, monotonic utility functions, but does the implication hold reversely as well?

• What do you denote by $K$ and $M$? – denesp Nov 30 '18 at 19:21
• @denesp K is the no. of firms and M is the total no. of goods – Shinjini Rana Nov 30 '18 at 23:48
• I see. So how do you get $p$? Is it any old $p$, or is it an equilibrium price vector? – denesp Dec 1 '18 at 9:15
• @denesp It is the eq. price vector. I should clarify most things in the doubt. What I really want to know is if all Pareto Optimal allocations are Competitive Equilibrium, for a general eq. model involving production, that is. I know that all Competitive allocations would be Pareto Optimal, given assumption of strictly increasing utility. What about the reverse? Does it hold? I know it doesn't for a general eq. model without production; Pareto Optimal allocations are NOT necessarily a subset of Competitive eq. – Shinjini Rana Dec 1 '18 at 13:29

## 1 Answer

You seem to be looking for the Second Welfare Theorem.

The Second Theorem states that out of all possible Pareto optimal outcomes one can achieve any particular one by enacting a lump-sum wealth redistribution and then letting the market take over.

• I'm not sure how the Second Welfare Theorem proves that all Pareto are Walrasian. That is, the set $C$ is equal to set $P$ (denoting set of competitive and Pareto allocations respectively). I'm sure I'm missing something, but how does Second Welfare Theorem ensure that each is a proper subset of the other, although from what I understand, not all Pareto allocations might be Competitive. If not, are there any specific conditions for the same? – Shinjini Rana Dec 1 '18 at 16:20
• From what you suggest, the condition is that there would be redistribution and transfers that would engineer the market such that the allocations which are competitively decided would be pareto. Is this right? – Shinjini Rana Dec 1 '18 at 16:23
• I suggest merely that what you are looking for is exactly the second welfare theorem. If you agree, you should probably read about it in advanced micro books like Mas-Colell, Winston, Green rather than Wikipedia. – denesp Dec 1 '18 at 19:53