Skip to content
Calculus II: Vector Calculus and Multivariable Analysis / Multivariate Optimisation

Derivation — Why Lagrange Multipliers Work

The Lagrange condition ∇f=λ∇g\nabla f = \lambda\nabla g is usually presented as a method without a proof: "at a constrained optimum, the gradients must be parallel." This note fills that gap with two connected proofs. The first uses the implicit function theorem to locally reduce the constrained problem to a single-variable problem, then applies the chain rule to derive the condition. The second is the shorter gradient-perpendicularity argument already sketched in the derivation lesson on steepest ascent. Together they also prove the shadow price formula λ=df∗/dc\lambda = df^*/dc: the Lagrange multiplier is the rate at which the optimal objective value changes when the constraint level shifts.
Learning objectives. By the end of this note you should be able to:
  • prove the Lagrange condition ∇f=λ∇g\nabla f = \lambda\nabla g via the implicit function theorem and chain rule;
  • prove the shadow price formula df∗(c)/dc=λ∗(c)df^*(c)/dc = \lambda^*(c);
  • state the regularity condition under which the method is valid (∇g≠0\nabla g \ne \mathbf{0});
  • identify when the method fails (constraint is degenerate at the optimum).

In this lesson

    1. The Implicit Function Theorem Setup
    1. Proof of the Lagrange Condition
    1. The Shadow Price Formula
    1. The nn-Variable Case
    1. When the Method Fails
    1. Lagrangian and Duality

Before you start

The full lesson, with worked examples and exercise solutions, is for members.

Read the full lesson

New to SelfMadeQuant.com? See membership options or explore the curriculum.