The first theorem of welfare economics states that "Every Walrasian Equilibrium (WE) is Pareto Efficient (PE)" - Microeconomic Theory Nicholson, Synder 11th ed. p477. But PE is quite a weak condition. Is WE a stronger condition or is does WE just mean any PE allocation?
I was specifically wondering; suppose we have an exchange economy with agents $N$, and there is a unique competitive equilibrium (WE) solution, $q^*$. Does this also maximize aggregate social welfare amongst possibly many PE outcomes? Where aggregate social welfare is defined as:
$$ SW(q^*) = \sum_i^N U_i(q_i^*) $$
To ask this question only makes sense if WE $\subset$ PE allocations, because if there is a single PE allocation then it must by definition maximize aggregate social welfare, because there is no other feasible solutions.
Any solution maximizing social welfare has to be PE, but not all PE solutions are SW maximizing. For example, at a given PE allocation agent $i$ could lose 1 util in exchange for agent $j$ gaining 2.