Envelope theorem

Remarks

For every $t$ one has $V(t) \ge f(x_0, t)$, with equality at $t = t_0$. So $t \mapsto V(t) - f(x_0, t)$ has a local minimum at the interior point $t_0$, and its gradient vanishes there. In words: to first order, the change of the optimal value equals the direct effect of the parameter, with the maximizer held fixed.

Example: $f(x, t) = tx - x^2$ on $\R \times \R$. The maximizer is $x = t/2$ and $V(t) = t^2/4$, so $V'(t) = t/2$. The theorem predicts $V'(t) = \partial_t f(x, t) = x = t/2$, which agrees.

The classical version assumes that $X$ is an open subset of $\R^n$, that $f$ is continuously differentiable, and that a maximizer $x^*(t)$ depends differentiably on $t$. Then the chain rule gives $\nabla V = \nabla_t f + (\nabla_x f) \, Dx^*$, and the second term vanishes because $\nabla_x f(x^*(t), t) = 0$. For a constrained problem, the same idea shows that the derivative of the optimal value with respect to a constraint level is the corresponding Lagrange multiplier. Differentiability of $V$ is a real hypothesis: for $X = \set{-1, 1}$ and $f(x, t) = xt$, the value $V(t) = \abs{t}$ is not differentiable at $0$.

Going deeper

For a student

The theorem is used throughout economics, for example to derive Hotelling's lemma and Roy's identity, and in optimization to compute sensitivities of optimal values. In optimal control and the calculus of variations, the same reasoning shows that the costate is the gradient of the value function, when the latter is differentiable. The version stated here, which needs no differentiability of the maximizer, is due to Paul Milgrom and Ilya Segal. When $V$ is not differentiable, one-sided derivatives of $V$ are given by Danskin's theorem under compactness assumptions.