Lagrange Multipliers

Optimising Under Constraint

Most economic decisions are constrained: maximise utility subject to a budget or cost. Lagrange multipliers solve these problems systematically.

The setup: maximise $f(x,y)$ subject to $g(x,y) = c$. Form the Lagrangian:

$$\mathcal{L}(x,y,\lambda) = f(x,y) - \lambda \bigl(g(x,y) - c\bigr)$$

First-order conditions require setting all partial derivatives to zero:

The shadow price $\lambda$ is the rate at which $f$ changes if the constraint relaxes by one unit: $\lambda \approx \frac{\Delta f^*}{\Delta c}$. Geometrically, level curves are tangent here.

The Shadow Price in Action

The multiplier $\lambda$ answers how much a constraint costs. In utility maximisation, the tangency condition is:

$$\frac{MU_x}{p_x} = \frac{MU_y}{p_y} = \lambda$$

This equalises marginal utility per euro spent. If $\lambda$ is large, the constraint is tight; if near zero, it is slack.

Common pitfall: Paying more than $\lambda$ to relax a constraint. If overtime costs 40 euro/hour but $\lambda = 25$ euro, expanding destroys value.

The Lagrange recipe:

Always report $\lambda$ alongside the optimum: it prices the constraint and drives business decisions.