Business I / Lagrange Multipliers

Practice question · Match the pairs

Match each economic problem to its Lagrange setup.

Utility maximisation
Cost minimisation
Portfolio optimisation
Shadow price

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

Hints

Every such problem has the same structure.
Identify what is being maximised and what limits it.

Show the 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.

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