 Lagrange multiplier as a marginal cost

The chosen quadratic objective function above gives more than an optimal allocation: its Lagrange multiplier predicts how much the best cost changes when the required total changes. For this example,This is an explicit derivative of a constrained value function and an instance of the envelope theorem. At $c=3$, one extra small unit of required output costs approximately $4$ units per unit of output, in the chosen normalized model. Increasing $c$ makes this equality-constrained minimization problem more expensive; it is not a gain from relaxing an upper resource limit. The sign and meaning depend on how the optimization constraint and optimization Lagrangian are written.^[ref]

For an actual finite change $\delta$, the cost difference is exactlyAt $\delta =0.1$, the linear approximation predicts $0.4$, while the exact increase is $61/150\approx 0.406667$. The extra $1/150$ comes from the squared term in this polynomial, rather than numerical error. A Lagrange multiplier gives a local sensitivity analysis; treating it as a constant price over a large change generally loses the change in the optimum itself.

For a smooth optimizer $z(c)$ of an equality problem $g(z)=c$, the short chain rule calculation explains the result: $\nabla g(z(c))\cdot z^{\prime }(c)=1$ and $\nabla f(z(c))=\lambda (c)\nabla g(z(c))$ imply $df(z(c))/dc=\lambda (c)$. The differentiability and stationarity assumptions matter: an optimizer can fail to be unique or vary smoothly. The envelope theorem records when this interpretation is justified.