<p>The social planner problem is 
\begin{align}
\max_{\lbrace x_i,y_i,x_j,y_j \rbrace } &amp;\theta_i U_i(x_i,y_i) +\theta_j U_i(x_j,y_j) \\
&amp; \text{s.t.}\\
&amp;x_i+x_j = \omega_i^x + \omega_j^x\\
&amp;y_i+y_j = \omega_i^y + \omega_j^y\\
\end{align}
where 
$\theta_i,\theta_j$ are weights.</p>

<p>As I understand it, with a proper choice of weights, one can turn the Social Planner's problem into </p>

<ol>
<li>A Lagrangian optimization problem that solves for a Pareto optimal allocation. </li>
<li>A Lagrangian optimization problem that solves for an allocation that is a Walrasian equilibrium. </li>
</ol>

<p>Is this accurate? If so, what constraints are there on the weights for each case? </p>

CC BY-SA 3.0
2015-10-23T23:52:06.293Z
Is there any particular reason why there are different wealths set aside for good x and good y for agents i and j?
Kitsune Cavalry
CC BY-SA 3.0
2015-10-24T06:42:40.150Z
@KitsuneCavalry I think these are the initial endowments for the Walrasian equilibrium.
Giskard
CC BY-SA 3.0
2015-10-24T06:44:45.123Z
This statement seems accurate (given the usual conditions on $U$) and you can arrive at the 1. conclusion by straightforward calculation, and at the 2. by thinking a little bit about the calculation used in the 1. case.
Giskard
CC BY-SA 3.0
2015-10-24T07:07:25.740Z
I'm trying to think if property one holds if the utility function for either agent is locally satiated. I think it does, but I can't work out the Lagrangian in my head.
Kitsune Cavalry
CC BY-SA 3.0
2015-10-26T10:36:17.830Z
If you work out the first order conditions, you can combine the conditions and obtain two sets of conditions, the interpersonal equity conditions and the Pareto optimality conditions. The Pareto optimality conditions can be derived no matter what weights you choose. For the latter question, I then recommend the second welfare theorem.
HRSE
<general-equilibrium>
Can the social planner problem be reduced to a Pareto problem or a Walrasian equilibrium problem?
Stan Shunpike
# Can the social planner problem be reduced to a Pareto problem or a Walrasian equilibrium problem?

(Asked by: username_0 on 2015-10-23)

The social planner problem is
\begin{align}
\max\_{\lbrace x\_i,y\_i,x\_j,y\_j \rbrace } &\theta\_i U\_i(x\_i,y\_i) +\theta\_j U\_i(x\_j,y\_j) \\
& \text{s.t.}\\
&x\_i+x\_j = \omega\_i^x + \omega\_j^x\\
&y\_i+y\_j = \omega\_i^y + \omega\_j^y\\
\end{align}
where
$\theta\_i,\theta\_j$ are weights.

As I understand it, with a proper choice of weights, one can turn the Social Planner's problem into

1. A Lagrangian optimization problem that solves for a Pareto optimal allocation.
2. A Lagrangian optimization problem that solves for an allocation that is a Walrasian equilibrium.

Is this accurate? If so, what constraints are there on the weights for each case?

**Tags:** general-equilibrium

> **Comments:**
> **username_1**: Is there any particular reason why there are different wealths set aside for good x and good y for agents i and j?  
> **username_2**: @username_1 I think these are the initial endowments for the Walrasian equilibrium.  
> **username_2**: This statement seems accurate (given the usual conditions on $U$) and you can arrive at the 1. conclusion by straightforward calculation, and at the 2. by thinking a little bit about the calculation used in the 1. case.  
> **username_1**: I'm trying to think if property one holds if the utility function for either agent is locally satiated. I think it does, but I can't work out the Lagrangian in my head.  
> **username_3**: If you work out the first order conditions, you can combine the conditions and obtain two sets of conditions, the interpersonal equity conditions and the Pareto optimality conditions. The Pareto optimality conditions can be derived no matter what weights you choose. For the latter question, I then recommend the second welfare theorem.  

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## 0 answers

