At least three different “envelope theorems” have proved useful for economic analysis. One applies to unconstrained optimization problems with parameterized objectives and unique solutions, a second to constrained, smooth concave maximization problems in which both the objective and constraint may be parameterized, and a third, which had not previously been given a general statement, to problems with parameterized objectives and any number of solutions. We state and prove the third theorem, generalize the first, and develop the precise relationship among the three.
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