The social welfare of a unique general competitive equilibrium in an exchange economy

Question

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.

Answer 1

Using the 'utility monster' critique of aggregate utility maximization it is pretty easy to show that gen. comp. eq. does not always maximize aggregate utility.

Imagine a pure exchange economy with two agents, $A$ and $B$ and two types of goods, $x_1$ and $x_2$. Agent $A$ has nothing whereas agent $B$ has one unit of each good. The agents respective utility functions are $$ U_A(x_1^A,x_2^A) = 2x_1^A + 2x_2^A \hskip 20pt U_B(x_1^B,x_2^B) = x_1^B + x_2^B. $$ Trivially the initial distribution is a competitive equilibrium and also trivially it does not maximize aggregate utility.

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EDIT:
I am not sure I understand @spinkus's comment, as obviously not every solution maximizes the aggregate utility, but I will address the other point: Not trading. That is not at all important for me, I just wanted to make the example trivial.

Suppose given some strictly monotonic utility functions $U_A,U_B$ and some initial endowments $w_A,w_B$ some allocation $a^*$ is a competitive equilibrium. If this allocation does not maximize aggregate utility we are done. If it does, consider the following monotonic transformation of $U_B$: $$ \hat{U}_B \triangleq c \cdot U_B $$ where $c$ is a positive number. In a pure exchange economy with $U_A,\hat{U}_B$, initial endowments $w_A,w_B$ the competitive equilibrium will again be $a^*$ as the preferences are unchanged. Yet even if $a^*$ maximized $U_A+U_B$, it clearly will not maximize $U_A + \hat{U}_B$ if $c$ is small enough.