Static optimisation problems

Static optimisation means choosing the best values of variables at one point in time, without dynamics. Unconstrained optimisation sets derivatives to zero; constrained optimisation — the usual economic case, since budgets and resources bind — uses the Lagrangian, which converts a constrained problem into an unconstrained one by pricing the constraint. Worked example: maximise utility U = xy subject to 2x + 4y = 100. The Lagrangian gives y = 2.lamda and x = 4.lamda (from first-order conditions), and the budget gives 2(4.lamda) + 4(2.lamda) = 16.lamda = 100, so lamda = 6.25, x = 25, y = 12.5.