Envelope theorem

In mathematics and economics, the envelope theorem is a major result about the differentiability properties of the value function of a parameterized optimization problem.^[1] As we change parameters of the objective, the envelope theorem shows that, in a certain sense, changes in the optimizer of the objective do not contribute to the change in the objective function. The envelope theorem is an important tool for comparative statics of optimization models.^[2]

The term envelope derives from describing the graph of the value function as the "upper envelope" of the graphs of the parameterized family of functions ${\displaystyle \left\{f\left(x,\cdot \right)\right\}_{x\in X}}$ that are optimized.