For a smooth constrained maximum subject to , suppose smooth optimizers and multipliers satisfy . Differentiating the constraint gives , and the chain rule gives . Thus a Lagrange multiplier measures the marginal value of relaxing its corresponding constraint. A global supporting Lagrangian duality representation implies the needed stationarity, but local stationarity and smoothness alone suffice for this derivative identity.
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The Envelope Theorem is a concept in economics, particularly in the fields of optimization and comparative statics. It describes how the value of an optimal objective function changes with respect to changes in parameters of the model. The fundamental idea is that when evaluating the impact of a change in parameters on the optimal value of the objective function, we can typically simplify the analysis by looking at the optimal solution without needing to find the explicit form of the solution again.