I need a little conceptual clarification. For a standard $N*K*M$ general equilibrium model, would an allocation, say, $y^k$ be Pareto Optimal if it does not solve $max(py^k)$? I understand that the competitive allocations would Pareto Optimal given strictly convex, monotonic utility functions, but does the implication hold reversely as well?
You seem to be looking for the Second Welfare Theorem.
The Second Theorem states that out of all possible Pareto optimal outcomes one can achieve any particular one by enacting a lump-sum wealth redistribution and then letting the market take over.