<p>Consider a <a href="http://web.stanford.edu/~jdlevin/Econ%20202/General%20Equilibrium.pdf" rel="nofollow noreferrer">pure exchange economy</a> with $N\geq2$ consumers, each of whom has <a href="https://en.wikipedia.org/wiki/Local_nonsatiation" rel="nofollow noreferrer">locally non-satiated</a> preferences. We know from the <a href="https://en.wikipedia.org/wiki/Fundamental_theorems_of_welfare_economics#Implications_of_the_First_Theorem" rel="nofollow noreferrer">First Welfare Theorem</a> that any <a href="https://en.wikipedia.org/wiki/Competitive_equilibrium" rel="nofollow noreferrer">Walrasian equilibrium</a> is <a href="https://en.wikipedia.org/wiki/Pareto_efficiency" rel="nofollow noreferrer">Pareto-efficient</a>. We index consumers by superscripts.</p>

<p>Say that consumer $i$ envies consumer $j$'s allocation if $i$ strictly prefers $j$'s allocation to their own:</p>

<p>$$ x^j \succ^i x^i $$</p>

<p>If these preferences have a utility representation, then this is equivalent to</p>

<p>$$ u^i (x^j) &gt; u^i(x^i) $$</p>

<p>It is an easy consequence of the First Welfare Theorem that in any Walrasian equilibrium, no two agents, regardless of income, envy each other's allocations. If they did, this would violate Pareto-efficiency.</p>

<p>Notice, however, that this is <strong>not</strong> the same as the allocation being <a href="https://en.wikipedia.org/wiki/Envy-freeness" rel="nofollow noreferrer">envy-free</a>.</p>

CC BY-SA 3.0
2016-12-09T14:58:27.350Z
2016-12-09T14:58:27.350Z
Theoretical Economist
<p>In a economy with an arbitrary number of agents whose preferences all satisfy the usual equilibrium conditions and with each starting with a random starting endownment, are there any two agents with the same income at equilibrium who might envy eachother?</p>

<p>Please notice that not all agents have the same income.</p>

CC BY-SA 3.0
2016-12-09T03:01:53.307Z
I think I need more detail about how you've specified the model to make sure, but surely if two agents envy each other's allocations, they would be willing to trade with each other?
Theoretical Economist
CC BY-SA 3.0
2016-12-09T03:49:14.690Z
Yeah I feel like if TWO people envy each other specifically then it's like...you're not at an equilibrium unless there's some nutty stuff going on rofl
Kitsune Cavalry
CC BY-SA 3.0
2016-12-09T04:27:53.477Z
@TheoreticalEconomist What I'm hazily trying to convey here is the idea of an Edgeworth (hyper)box where agents have well-behaved utility functions (e.g. continuous, monotonic, differentiable and concave). But yeah, now that you've mentioned, they would gain from trade. I've missed that ball pretty hard.
ejQhZ
CC BY-SA 3.0
2016-12-09T14:59:29.823Z
@ejQhZ It's easy to miss obvious things with these models. I've written up an answer in more detail below for anyone who might be reading this and is interested.
Theoretical Economist
<microeconomics><fairness>
In a trading economy with many agents, can two agents with the same income envy eachother?
ejQhZ
# In a trading economy with many agents, can two agents with the same income envy eachother?

(Asked by: username_0 on 2016-12-09)

In a economy with an arbitrary number of agents whose preferences all satisfy the usual equilibrium conditions and with each starting with a random starting endownment, are there any two agents with the same income at equilibrium who might envy eachother?

Please notice that not all agents have the same income.

**Tags:** microeconomics, fairness
> **Comments:**
> **username_1**: I think I need more detail about how you've specified the model to make sure, but surely if two agents envy each other's allocations, they would be willing to trade with each other?  
> **username_2**: Yeah I feel like if TWO people envy each other specifically then it's like...you're not at an equilibrium unless there's some nutty stuff going on rofl  
> **username_0**: @TheoreticalEconomist What I'm hazily trying to convey here is the idea of an Edgeworth (hyper)box where agents have well-behaved utility functions (e.g. continuous, monotonic, differentiable and concave). But yeah, now that you've mentioned, they would gain from trade. I've missed that ball pretty hard.  
> **username_1**: @ejQhZ It's easy to miss obvious things with these models. I've written up an answer in more detail below for anyone who might be reading this and is interested.  

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## 1 answers

### Answer by username_1 on 2016-12-09. Score: 0 | Accepted answer

Consider a [pure exchange economy](http://web.stanford.edu/~jdlevin/Econ%20202/General%20Equilibrium.pdf) with $N\geq2$ consumers, each of whom has [locally non-satiated](https://en.wikipedia.org/wiki/Local_nonsatiation) preferences. We know from the [First Welfare Theorem](https://en.wikipedia.org/wiki/Fundamental_theorems_of_welfare_economics#Implications_of_the_First_Theorem) that any [Walrasian equilibrium](https://en.wikipedia.org/wiki/Competitive_equilibrium) is [Pareto-efficient](https://en.wikipedia.org/wiki/Pareto_efficiency). We index consumers by superscripts.

Say that consumer $i$ envies consumer $j$'s allocation if $i$ strictly prefers $j$'s allocation to their own:

$$ x^j \succ^i x^i $$

If these preferences have a utility representation, then this is equivalent to

$$ u^i (x^j) > u^i(x^i) $$

It is an easy consequence of the First Welfare Theorem that in any Walrasian equilibrium, no two agents, regardless of income, envy each other's allocations. If they did, this would violate Pareto-efficiency.

Notice, however, that this is **not** the same as the allocation being [envy-free](https://en.wikipedia.org/wiki/Envy-freeness).


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