= Envelope theorem
{title2=$\nabla_b\phi(b)=\lambda(b)$}
{wiki}
For a smooth constrained maximum $\phi(b)=f(\bar x(b))$ subject to $g(\bar x(b))=b$, suppose smooth optimizers and multipliers satisfy $\nabla f(\bar x)=Dg(\bar x)^T\lambda$. Differentiating the constraint gives $Dg(\bar x)D_b\bar x=I$, and the chain rule gives $D_b\phi=\lambda^T$. Thus a <Lagrange multiplier> measures the marginal value of relaxing its corresponding constraint. A global supporting <Lagrangian duality> representation implies the needed stationarity, but local stationarity and smoothness alone suffice for this derivative identity.
Back to article page
