# 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 $M$? – Giskard 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? – Giskard 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

• 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