Lagrangian Multipliers
Constrained optimization's workhorse, and why the multiplier is a shadow price.
Definition
The method for optimization under equality constraints: bundle objective and constraint into one function; at the optimum the constraint's multipliermultiplierThe amount total output changes per dollar of initial spending change, powered by respending: 1/(1−MPC) in the simplest case. prices its scarcity.
Key equation
ℒ = f(x) + λ(c − g(x)); FOCs: ∇f = λ∇g
The intuition
At the best feasible point, the objective's gradient must be parallel to the constraint's, otherwise you could slide along the constraint and improve. λ is the shadow price: the objective's gain from one more unit of the resource.
Exam tip
Always interpret λ economically (marginal utility of income, marginal costmarginal costThe cost of producing one more unit. of the quota), that's where the marks are.
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