# Lagrange Multipliers

Business I · Mathematics II · https://tryals.app/learn/business-i/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:

| Derivative | Condition | Meaning |
| --- | --- | --- |
| $\partial \mathcal{L} / \partial x$ | $f_x - \lambda g_x = 0$ | Marginal return per cost unit |
| $\partial \mathcal{L} / \partial y$ | $f_y - \lambda g_y = 0$ | Equalised across variables |
| $\partial \mathcal{L} / \partial \lambda$ | $g(x,y) = c$ | Enforces the constraint |

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**:

| Step | Action |
| --- | --- |
| 1 | Write $\mathcal{L} = f - \lambda(g - c)$ |
| 2 | Set $f_x = \lambda g_x$ and $f_y = \lambda g_y$ |
| 3 | Set $\partial\mathcal{L}/\partial\lambda = 0$ |
| 4 | Solve the system; read $\lambda$ as the bonus |

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

## Practice questions

7 of this lesson's 13 practice questions, with answers. The full set is in the app.

### 1. The Lagrange multiplier $\lambda$ is sometimes dismissed as a bookkeeping device with no meaning. What does it actually measure?

A. The rate at which the objective function is changing along the constraint
B. The distance between the objective function and the constraint at the optimum
C. Nothing measurable: it is genuinely an artefact of the method
D. The shadow price: how much the optimum improves per unit

**Answer:** D. The shadow price: how much the optimum improves per unit

**Why:** $\lambda$ is the value of relaxing the constraint by one unit. This makes it directly usable: it prices an extra hour of machine time or an extra euro of budget without re-solving the problem.

Page: https://tryals.app/practice/business-i/lagrange-multipliers/the-lagrange-multiplier-is-sometimes-dismissed-as-a-bookkeeping

### 2. The Lagrange multiplier λ has an economic meaning at the optimum. What is it?

A. The shadow price: the gain from relaxing the constraint by one
B. The exchange rate between the two goods in the objective function
C. The rate at which the constraint is being violated at the optimum
D. A purely technical quantity introduced to make the algebra work

**Answer:** A. The shadow price: the gain from relaxing the constraint by one

**Why:** λ is the derivative of the optimal value with respect to the constraint level, so it prices the constraint itself, which turns 'is the extra machine worth buying?' into a number you already computed. Treating it as algebraic bookkeeping discards the most useful output of the method.

Page: https://tryals.app/practice/business-i/lagrange-multipliers/the-lagrange-multiplier-has-an-economic-meaning-at-the-optimum

### 3. At the consumer's optimum, $MU_x / p_x = MU_y / p_y$. What does this say?

A. Prices must be equal
B. The consumer spends equally on both goods
C. Marginal utility per euro is equalised
D. Marginal utilities must be equal

**Answer:** C. Marginal utility per euro is equalised

**Why:** Equalised bang-for-buck: if $MU_x / p_x > MU_y / p_y$, shifting a euro from $y$ to $x$ raises utility, the optimum is where no reallocation helps.

Page: https://tryals.app/practice/business-i/lagrange-multipliers/at-the-consumers-optimum-mux-px-muy-py-what-does-this-say

### 4. Which statements about $\lambda$ are correct?

A. It measures how much the optimum improves per unit of constraint relaxation
B. It is called the shadow price
C. It has units of objective per unit of constraint
D. It is always positive

**Answer:** A. It measures how much the optimum improves per unit of constraint relaxation; B. It is called the shadow price; C. It has units of objective per unit of constraint

**Why:** $\lambda$ is the shadow price with units of objective/constraint. It can be negative (in minimisation problems with reversed constraint direction).

Page: https://tryals.app/practice/business-i/lagrange-multipliers/which-statements-about-are-correct

### 5. At the constrained optimum, the level curves of the objective function are tangent to the constraint curve.

**Answer:** True

**Why:** Tangency means the gradients are proportional: $\nabla f = \lambda \nabla g$. Any movement along the constraint away from this point would worsen the objective.

Page: https://tryals.app/practice/business-i/lagrange-multipliers/at-the-constrained-optimum-the-level-curves-of-the-objective

### 6. A factory has $\lambda = 8$ for its capacity constraint. Management can add 10 units of capacity. **Estimate the output gain.**

**Answer:** 80 (within ±5)

**Why:** $8 \times 10 = 80$ units of output. The shadow price converts constraint relaxation directly into objective improvement.

Page: https://tryals.app/practice/business-i/lagrange-multipliers/a-factory-has-8-for-its-capacity-constraint-management-can-add

### 7. Match each economic problem to its Lagrange setup.

**Answer:**

- Utility maximisation → Maximise $U(x,y)$ subject to budget $p_x x + p_y y = M$
- Cost minimisation → Minimise $wL + rK$ subject to output $Q(K,L) = Q_0$
- Portfolio optimisation → Minimise variance subject to target return
- Shadow price → $\lambda$: the shadow price of the binding constraint

**Why:** Every constrained optimisation problem has the same structure, only the objective and constraint change.

Page: https://tryals.app/practice/business-i/lagrange-multipliers/match-each-economic-problem-to-its-lagrange-setup