# In a trading economy with many agents, can two agents with the same income envy eachother?

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.

• 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? Dec 9, 2016 at 3:01
• 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 Dec 9, 2016 at 3:49
• @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. Dec 9, 2016 at 4:27
• @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. Dec 9, 2016 at 14:59

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.