Envelope Theorem
A curve in a two dimensional space is best represented by the parametric equations like x(c) and y(c). The family of curves can be represented in the form where c is the parameter. Generally, the envelope theorem involves one parameter but there can be more than one parameter involved as well.
The envelope of a family of curves g(x,y,c) = 0 is a curve such that at each point on the curve there is some member of the family that touches that particular point tangentially. This forms a curve or surface that is tangential to every curve in the family of curves forming an envelope.
Consider an arbitrary maximization (or minimization) problem where the objective function depends on some parameters :
The function is the problem's optimal-value function — it gives the maximized (or minimized) value of the objective function as a function of its parameters .
Let be the (arg max) value of, expressed in terms of the parameters, that solves the optimization problem, so that . The envelope theorem tells us how changes as a parameter changes, namely:
That is, the derivative of with respect to is given by the partial derivative of with respect to, holding fixed, and then evaluating at the optimal choice .
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