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.
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

$$ u^i (x^j) &gt; u^i(x^i) $$

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.