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Interpret a beneficial trade as weakly improving both participants and strictly improving at least one; otherwise the identity trade would always remain available. Put $g_j=\nabla U_j(y_j)$. For a globally strictly increasing concave function, every component of its gradient is strictly positive: concavity makes a coordinate derivative nonincreasing along that coordinate, and monotonicity makes it nonnegative. If it vanished at a finite point, it would vanish farther along the coordinate, contradicting strict increase on the whole real line.

To construct a pairwise improvement from nonparallel utility gradients, suppose two positive gradients $g,h$ are not proportional. Strict Cauchy-Schwarz inequality givesChoose $t$ between these bounds and put $\delta =g-th$. Then $g\cdot \delta >0$ and $h\cdot \delta <0$. For sufficiently small $\epsilon >0$, the feasible transfer $y_i\mapsto y_i+\epsilon \delta$, $y_j\mapsto y_j-\epsilon \delta$ strictly improves both utilities by differentiability. This contradicts the absence of beneficial pairwise trades.

Hence all gradients are proportional. Choose $v=g_1$; positivity makes each proportionality scalar positive:Now take $\boxed{a_j=1/{\lambda }_j}$. The concave supporting-tangent inequality gives, for any other allocation $(z_j)$ with the same total endowment,Thus the final allocations globally maximize the social planner's weighted objective. They need not be unique when utilities are not strictly concave, but the required positive supporting weights exist.