Lagrange Multipliers

Optimising Under Constraint

Most economic decisions are constrained: maximise utility subject to a budget, minimise cost subject to an output target, maximise profit subject to a production capacity. Lagrange multipliers solve these problems systematically.

The setup: maximise (or minimise) $f(x,y)$ subject to the constraint $g(x,y) = c$.

Form the Lagrangian:

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

First-order conditions (set all partials to zero):

$$\frac{\partial \mathcal{L}}{\partial x} = f_x - \lambda g_x = 0$$ $$\frac{\partial \mathcal{L}}{\partial y} = f_y - \lambda g_y = 0$$ $$\frac{\partial \mathcal{L}}{\partial \lambda} = -(g(x,y) - c) = 0$$

The first two conditions say $f_x / g_x = f_y / g_y = \lambda$ — at the optimum, the marginal rate of return per unit of constraint cost is equalised across all choice variables. The third condition enforces the constraint itself.

The economic meaning of $\lambda$: it is the shadow price of the constraint — the rate at which the optimal value of $f$ changes if the constraint is relaxed by one unit. If $\lambda = 5$ and the constraint is a budget, then one more euro of budget buys approximately 5 units of the objective. This is the single most useful number in applied optimisation.

$$\lambda \approx \frac{\Delta f^*}{\Delta c}$$

Geometric intuition: at the constrained optimum, the gradient of $f$ is proportional to the gradient of $g$ — the objective's level curves are tangent to the constraint curve. Moving along the constraint in either direction reduces the objective; you're at the best feasible point.

The Lagrange recipe

Tip: The geometric heart: at the constrained optimum, the level curve of $f$ and the constraint curve are tangent — their gradients align, and $\lambda$ is precisely the proportionality factor between them.

The Shadow Price in Action

The multiplier $\lambda$ is not just a mathematical by-product — it is the answer to the most important practical question in constrained optimisation: how much is the constraint costing me?

Example: a firm maximises output $Q(K,L)$ subject to a cost constraint $wL + rK = C$. The Lagrangian gives $\lambda$ as the marginal product of budget — one more euro of spending buys $\lambda$ units of output. If $\lambda$ is large, the firm is severely constrained (it has high-return investments it can't afford); if $\lambda$ is near zero, the constraint isn't binding much (extra budget would add little).

$$\lambda_{\text{large}} \Rightarrow \text{tight constraint, relaxation valuable}$$ $$\lambda_{\text{small}} \Rightarrow \text{slack constraint, relaxation cheap}$$

Utility maximisation: a consumer maximises $U(x,y)$ subject to $p_x x + p_y y = M$. The first-order conditions give the tangency condition:

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

This says: at the optimum, the marginal utility per euro spent is equalised across all goods — and that common ratio is $\lambda$, the marginal utility of income. If you equalise bang-for-buck across goods, you've maximised total bang.

In practice: linear programming, cost-benefit analysis, engineering design, and portfolio optimisation all use shadow prices. When a government asks "how much is this environmental regulation costing us in GDP?", the answer is a Lagrange multiplier — the shadow price of the constraint.

Tip: Always report $\lambda$ alongside the optimum — it prices the constraint. In a budget problem it answers "what would one more euro buy?"; in a capacity problem, "what is one more machine-hour worth?" That number is the business case for relaxing the constraint.

Common pitfall: Paying more than $\lambda$ to relax a constraint. If overtime capacity costs 40€/hour but the shadow price is 25€/hour, the expansion destroys value — the multiplier is the ceiling on what relaxation is worth.