A note on Lagrange multipliers in the multiple constraint case

The well known reciprocal relationship among the Lagrange multiplier of the utility maximization problem subject to the budget constraint and the associated cost minimization problem has been recently extended to the multiple constraint case. A ‘dual’ minimization problem is attached to every constraint of the primal with the corresponding multipliers satisfying a reciprocal relationship. In general, the optimal solutions of the primal and dual problems are not the same. We prove that a maximum of the primal problem, that satisfies the second order sufficient conditions, is an extreme point of every dual problem. Furthermore, only when all the primal multipliers are strictly positive, the primal maximum is a minimum for all the dual problems.