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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.

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    $\begingroup$ 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? $\endgroup$ – Theoretical Economist Dec 9 '16 at 3:01
  • $\begingroup$ 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 $\endgroup$ – Kitsune Cavalry Dec 9 '16 at 3:49
  • $\begingroup$ @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. $\endgroup$ – ejQhZ Dec 9 '16 at 4:27
  • $\begingroup$ @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. $\endgroup$ – Theoretical Economist Dec 9 '16 at 14:59
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Consider a pure exchange economy with $N\geq2$ consumers, each of whom has locally non-satiated preferences. We know from the First Welfare Theorem that any Walrasian equilibrium is Pareto-efficient. 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.

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