<p>We have two Consumers 1 and 2, and two goods 1 and 2 pure exchange economy. The following utility functions can be used to represent their preferences:</p>

<ul>
<li>$u_1(x_{11}, x_{12}) = x_{11}$</li>
<li>$u_2(x_{21}, x_{22}) = x_{22}$</li>
</ul>

<p>Equilibrium price vector $(p_1, p_2=1)$ and allocation $((x_{11}, x_{12}), (x_{21}, x_{22}))$ satisfy the following:</p>

<p>Optimality Conditions (Allocation must solve the utility maximization problem of the two consumers, i.e. it must lie on their demand functions)</p>

<ul>
<li>$(x_{11}, x_{12}) = \left(\frac{50p_1 + 50}{p_1}, 0\right) $</li>
<li>$(x_{21}, x_{22}) = \left(0, 50p_1 + 50\right)$</li>
</ul>

<p>Market Clearing Conditions</p>

<ul>
<li>$x_{11} + x_{21} = 100$</li>
<li>$x_{12} + x_{22} = 100$</li>
</ul>

<p>Solving the above gives price vector $(p_1, p_2) = (1, 1)$ that supports the allocation $((x_{11}, x_{12}), (x_{21}, x_{22})) = ((100, 0), (0, 100))$ in equilibrium. This allocation is the only competitive equilibrium. It is also the only Pareto efficient allocation in this economy.</p>

CC BY-SA 3.0
2017-03-20T02:25:41.663Z
2017-03-20T02:25:41.663Z
Amit
<p>There are 100 units of good 1 and good 2 in a economy. Consumer 1 and consumer 2 have 50 units of each good. 
Consumer 1 only wants good 1 whereas consumer 2 only wants good 2.</p>

<p>Note: Neither of these are lexicographic preferences.</p>

<p><strong>Question</strong>: Find Walrasian equilibria</p>

<p>Here is what I have reached:</p>

<p><a href="https://i.sstatic.net/0b2Lk.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/0b2Lk.png" alt="enter image description here"></a></p>

<p>Can someone explain why red lines are WE..and why not F and G?
And P is Pareto optimal I understand, but is it Walrasian equilibrium as well?</p>

<p>Is every Walrasian equilibrium Pareto optimal and every Pareto optimal point Walrasian equilibrium?</p>

<pareto-efficiency><walrasian>
Pareto optimal and Walrasian equilibrium
HARSHA VARDHAN
# Pareto optimal and Walrasian equilibrium

(Asked by: username_0 on 2017-03-19)

There are 100 units of good 1 and good 2 in a economy. Consumer 1 and consumer 2 have 50 units of each good.
Consumer 1 only wants good 1 whereas consumer 2 only wants good 2.

Note: Neither of these are lexicographic preferences.

**Question**: Find Walrasian equilibria

Here is what I have reached:

[![enter image description here](https://i.sstatic.net/0b2Lk.png)](https://i.sstatic.net/0b2Lk.png)

Can someone explain why red lines are WE..and why not F and G?
And P is Pareto optimal I understand, but is it Walrasian equilibrium as well?

Is every Walrasian equilibrium Pareto optimal and every Pareto optimal point Walrasian equilibrium?

**Tags:** pareto-efficiency, walrasian


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## 1 answers

### Answer by username_1 on 2017-03-20. Score: 4

We have two Consumers 1 and 2, and two goods 1 and 2 pure exchange economy. The following utility functions can be used to represent their preferences:

* $u\_1(x\_{11}, x\_{12}) = x\_{11}$
* $u\_2(x\_{21}, x\_{22}) = x\_{22}$

Equilibrium price vector $(p\_1, p\_2=1)$ and allocation $((x\_{11}, x\_{12}), (x\_{21}, x\_{22}))$ satisfy the following:

Optimality Conditions (Allocation must solve the utility maximization problem of the two consumers, i.e. it must lie on their demand functions)

* $(x\_{11}, x\_{12}) = \left(\frac{50p\_1 + 50}{p\_1}, 0\right) $
* $(x\_{21}, x\_{22}) = \left(0, 50p\_1 + 50\right)$

Market Clearing Conditions

* $x\_{11} + x\_{21} = 100$
* $x\_{12} + x\_{22} = 100$

Solving the above gives price vector $(p\_1, p\_2) = (1, 1)$ that supports the allocation $((x\_{11}, x\_{12}), (x\_{21}, x\_{22})) = ((100, 0), (0, 100))$ in equilibrium. This allocation is the only competitive equilibrium. It is also the only Pareto efficient allocation in this economy.


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