Consumption Externalities in partial equilibrium analysis
Without any constraint the social planer would set $x_B \rightarrow \infty$, and not solve the &quot;first order condition&quot; given by $x_B+1/x_B=0$ only valid for an inner solution. The centralized problem is not comparable with the budget constrained decentralized problems you solved. (By the way why should $m$ be the same for both consumers?)