<p>You seem to be looking for the <a href="https://en.wikipedia.org/wiki/Fundamental_theorems_of_welfare_economics" rel="nofollow noreferrer">Second Welfare Theorem</a>.</p>

<blockquote>
  <p>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.</p>
</blockquote>

CC BY-SA 4.0
2018-12-01T16:20:43.243Z
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?
S.Rana
CC BY-SA 4.0
2018-12-01T16:23:36.820Z
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?
S.Rana
CC BY-SA 4.0
2018-12-01T19:53:50.267Z
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.
Giskard
CC BY-SA 4.0
2018-12-01T14:47:59.860Z
2018-12-01T14:47:59.860Z
Giskard
<p>I need a little conceptual clarification. 
For a standard <span class="math-container">$N*K*M$</span> general equilibrium model, would an allocation, say, <span class="math-container">$y^k$</span> be Pareto Optimal if it does not solve <span class="math-container">$max(py^k)$</span>? 
I understand that the competitive allocations would Pareto Optimal given strictly convex, monotonic utility functions, but does the implication hold reversely as well? </p>

CC BY-SA 4.0
2018-11-30T19:21:10.290Z
What do you denote by $K$ and $M$?
Giskard
CC BY-SA 4.0
2018-11-30T23:48:43.810Z
@denesp K is the no. of firms and M is the total no. of goods
S.Rana
CC BY-SA 4.0
2018-12-01T09:15:16.857Z
I see. So how do you get $p$? Is it any old $p$, or is it an equilibrium price vector?
Giskard
CC BY-SA 4.0
2018-12-01T13:29:53.837Z
@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.
S.Rana
CC BY-SA 4.0
2020-04-24T18:18:45.907Z
The definition of Pareto optimality should not depend on the equilibrium prices. You can easily build examples with a unique equilibrium but many different Pareto efficient allocations. Maybe your question is whether for any Pareto efficient allocation there exists a vector of prices such that...
brunosalcedo
<microeconomics><production-function><self-study><general-equilibrium><walrasian>
General Equilibrium Involving Production
S.Rana
# General Equilibrium Involving Production

(Asked by: username_0 on 2018-11-30)

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?

**Tags:** microeconomics, production-function, self-study, general-equilibrium, walrasian

> **Comments:**
> **username_1**: What do you denote by $K$ and $M$?  
> **username_0**: @denesp K is the no. of firms and M is the total no. of goods  
> **username_1**: I see. So how do you get $p$? Is it any old $p$, or is it an equilibrium price vector?  
> **username_0**: @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.  
> **username_2**: The definition of Pareto optimality should not depend on the equilibrium prices. You can easily build examples with a unique equilibrium but many different Pareto efficient allocations. Maybe your question is whether for any Pareto efficient allocation there exists a vector of prices such that...  

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## 1 answers

### Answer by username_1 on 2018-12-01. Score: 1

You seem to be looking for the [Second Welfare Theorem](https://en.wikipedia.org/wiki/Fundamental_theorems_of_welfare_economics).

> 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.

> **Comments:**
> **username_0**: 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?  
> **username_0**: 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?  
> **username_1**: 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.  

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