Lagrange Multipliers Intro

Business Calculus · Axiom Academy

LESSON Introduction to Lagrange Multipliers Optimizing with constraints using calculus In real business situations, we rarely have unlimited resources. We want to maximize profit subject to a budget constraint, or minimize cost while meeting a production target. This is constrained optimization . Maximize f(x, y) subject to g(x, y) = c Why Gradients Must Be Parallel At a constrained maximum (or minimum), the gradient of the objective function must be parallel to the gradient of the constraint. Why? If they weren't parallel, you could move along the constraint curve in a direction that increases (or decreases) f—meaning you haven't found the optimum yet! At the optimum, the gradients are parallel (λ is the Lagrange multiplier) The constraint (blue circle) is tangent to a level curve of f (green). At this point, ∇f and ∇g point in the same direction. L(x, y, λ) = f(x, y) - λ(g(x, y) - c) This gives a system of three equations in three unknowns Setting the partial derivatives to zero gives us: Equation 1: Marginal condition for x Equation 2: Marginal condition for y Equation 3: The constraint itself (must be satisfied) The Lagrange multiplier λ has a powerful economic interpretation: λ = the rate of change of the optimal value with respect to the constraint constant In other words: if you could relax the constraint by one unit, how much would the objective improve? Budget constraint: λ = marginal value of an extra dollar of budget

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