📜 Envelope Theorem
When there is a parameter in the optimization problem, how does the value function (the value of \(f\) at the optimum) depend on it? Let's start with the simplest case: Unconstrained optimization:
\textbf{Theorem 184} \(f: U \times I \rightarrow \mathbb{R}\) where \(U \subset \mathbb{R}^n\) open and \(I \subset \mathbb{R}\) interval is \(C^1\):
\[f(x, q)
\]
Suppose that for each \(q\), there is a solution \(x(q)\). If \(V(q) = f(x^*(q), q) = \max_{x \in \mathbb{R}^n} f(x, q)\), Suppose that \(q \rightarrow x^*(q)\) is of class \(C^1\), then:
\[\frac{d V(q)}{dq} = \frac{\partial f}{\partial q}(x^*(q), q)
\]
