<p>Hint:</p>

<blockquote>
  <p><strong>Second Welfare Theorem</strong>: Suppose that for every individual $i = 1,...,N$ the utility function $U^i(c_i)$ is continuous, locally non-satiated, and quasi-concave. If $\left\{c_1^P,...,c_N^P\right\}$ represents a Pareto optimal allocation in which $c_i^P &gt;&gt; 0$ for every $i = 1,...,N$, then there exists a set of prices, $p^E \geq 0$ and $p^E \neq 0$ and a set of transfer payments, $\left\{\tau_1,...,\tau_N\right\}$ such that:</p>
  
  <ul>
  <li>$\left\{c_1^P,...,c_N^P\right\}$ is feasible</li>
  </ul>
  
  <p>$$\sum^N_{i=1}c_i^P \leq \sum^N_{i=1}\omega_i^P$$</p>
  
  <ul>
  <li>For every individual $i = 1,...,N$, the consumption bundle, $c_i^P$, solves:</li>
  </ul>
  
  <p>$$\max_{c_i} U^i(c_i) \ s.t. \ p^E \cdot c_i = p^E \cdot \omega^i - \tau_i$$</p>
  
  <ul>
  <li>and $\sum^n_{i=1}\tau_i = 0$</li>
  </ul>
</blockquote>

<p>Or in other words, for any efficient (Pareto) allocation, there exists a price vector such that the price vector and optimal bundle represent a quasi-equilibrium with transfers.</p>

<p>Also note that you need preferences to be convex and locally non-satiated and the production set must be convex. So now what happens when $U_i(c_i) = \min \left\{\frac{c_1^P}{\alpha_1},...,\frac{c_N^P}{\alpha_N}\right\}$? It is indeed continuous, locally non-satiated and quasi-concave.</p>

CC BY-SA 3.0
2015-10-17T17:41:03.940Z
2015-10-18T03:52:38.990Z
2015-10-18T03:52:38.990Z
Kitsune Cavalry
<p>Does the 2nd welfare theorem hold with Leontief preferences?  If not, which of the assumptions does not hold?</p>

<preferences><welfare><leontief>
Leontief preferences and 2nd welfare theorem
Peter
# Leontief preferences and 2nd welfare theorem

(Asked by: username_0 on 2015-10-17)

Does the 2nd welfare theorem hold with Leontief preferences? If not, which of the assumptions does not hold?

**Tags:** preferences, welfare, leontief

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## 1 answers

### Answer by username_1 on 2015-10-17. Score: 4

Hint:

> **Second Welfare Theorem**: Suppose that for every individual $i = 1,...,N$ the utility function $U^i(c\_i)$ is continuous, locally non-satiated, and quasi-concave. If $\left\{c\_1^P,...,c\_N^P\right\}$ represents a Pareto optimal allocation in which $c\_i^P >> 0$ for every $i = 1,...,N$, then there exists a set of prices, $p^E \geq 0$ and $p^E \neq 0$ and a set of transfer payments, $\left\{\tau\_1,...,\tau\_N\right\}$ such that:
>
> * $\left\{c\_1^P,...,c\_N^P\right\}$ is feasible
>
> $$\sum^N\_{i=1}c\_i^P \leq \sum^N\_{i=1}\omega\_i^P$$
>
> * For every individual $i = 1,...,N$, the consumption bundle, $c\_i^P$, solves:
>
> $$\max\_{c\_i} U^i(c\_i) \ s.t. \ p^E \cdot c\_i = p^E \cdot \omega^i - \tau\_i$$
>
> * and $\sum^n\_{i=1}\tau\_i = 0$

Or in other words, for any efficient (Pareto) allocation, there exists a price vector such that the price vector and optimal bundle represent a quasi-equilibrium with transfers.

Also note that you need preferences to be convex and locally non-satiated and the production set must be convex. So now what happens when $U\_i(c\_i) = \min \left\{\frac{c\_1^P}{\alpha\_1},...,\frac{c\_N^P}{\alpha\_N}\right\}$? It is indeed continuous, locally non-satiated and quasi-concave.


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