ECON5005 · Quantitative Tools for Economics
Constrained Optimisation
Constrained optimisation is the heart of Week 7 in ECON5005 Quantitative Tools for Economics at the University of Sydney: how to maximise an objective — utility, output, profit — when a budget or resource limit ties the choice variables together. This chapter builds the three routes to the same answer (substitution, tangency, and the Lagrangian), extends them to inequality constraints via the Kuhn–Tucker conditions, and reads the multiplier as a shadow price — the value of relaxing the constraint by one unit.
What this chapter covers
- 01Set up a constrained problem: objective, choice variables, and an equality or inequality constraint
- 02Solve by substitution: rearrange the constraint, substitute, then optimise in one variable
- 03Apply the tangency condition MRS = price ratio, equivalently equal marginal utility per dollar
- 04Form the Lagrangian L = f + lambda(c - g) and solve its three first-order conditions
- 05See why tangency and the Lagrangian are the same condition: grad f = lambda grad g
- 06Interpret the multiplier lambda* as the shadow price, the extra objective per unit of relaxed constraint
- 07Handle inequality constraints with Kuhn-Tucker: stationarity, complementary slackness, feasibility
- 08Test whether a <= constraint binds or is slack (lambda = 0), and reject infeasible bundles
Utility maximisation by tangency and the Lagrangian, with the shadow price
- +1Marginal utilities: MU_x = dU/dx = y and MU_y = dU/dy = x.
- +2Tangency: MRS = MU_x/MU_y = y/x set equal to p_x/p_y = 2/4 = 1/2, so y/x = 1/2, i.e. x = 2y.
- +2Impose the budget line (the second equation): 2(2y) + 4y = 8y = 40, so y* = 5 and x* = 2(5) = 10.
- +2Lagrangian: L = xy + lambda(40 - 2x - 4y). FOCs: L_x = y - 2lambda = 0, L_y = x - 4lambda = 0, L_lambda = 40 - 2x - 4y = 0.
- +1From the first two FOCs, lambda = y/2 = x/4, giving x = 2y again; with the budget line x* = 10, y* = 5 and lambda* = y*/2 = 2.5.
- +1Utility attained: U* = x*y* = 10 x 5 = 50.
- +1Shadow price: lambda* = 2.5, so an extra $1 of income raises maximum utility by about 2.5 units. Check via the value function V(M) = M^2/(4 p_x p_y) = M^2/32, so dV/dM = 2M/32 = 40/16 = 2.5. Confirmed.
Key terms
- Constrained optimisation
- Maximising (or minimising) an objective f(x, y) subject to a constraint g(x, y) = c (or g(x, y) <= c) that links the choice variables, such as a budget line or a resource limit.
- Substitution method
- Solve the constraint for one variable, substitute it into the objective, then optimise the result as an ordinary single-variable problem using the first- and second-order conditions.
- Tangency condition
- At a constrained optimum the objective's level curve is tangent to the constraint. For utility maximisation this is MRS = MU_1/MU_2 = p_1/p_2, equivalently MU_1/p_1 = MU_2/p_2 (equal marginal utility per dollar).
- Lagrangian
- The function L = f(x, y) + lambda(c - g(x, y)) whose three first-order conditions (L_x = 0, L_y = 0, L_lambda = 0) locate the constrained optimum for any smooth constraint.
- Lagrange multiplier (lambda)
- The auxiliary variable in the Lagrangian. At the optimum grad f = lambda grad g, so lambda measures how tightly the constraint pulls the objective away from its free optimum.
- Shadow price
- The value of the multiplier at the optimum, lambda* = d V*/dc, where V*(c) is the maximised objective. It is the extra objective gained per one-unit relaxation of the constraint — e.g. utility per extra dollar of budget.
- Kuhn-Tucker conditions
- The rules for an inequality constraint g(x) <= c: stationarity (L_xi = 0), complementary slackness (lambda >= 0 and lambda(g - c) = 0), and feasibility (g <= c).
- Complementary slackness
- Either the constraint binds (g = c) or its multiplier is zero (lambda = 0). A slack constraint has a zero shadow price, because spare resource is worth nothing at the margin.
Constrained Optimisation FAQ
Do the tangency method and the Lagrangian give the same answer?
Yes — they are the same condition written two ways. Dividing the first Lagrangian first-order condition by the second gives f_x/f_y = g_x/g_y, which is exactly MRS = price ratio (the tangency condition). Geometrically both say grad f and grad g point along the same line at the optimum, so the objective's level curve and the constraint share a tangent. The Lagrangian is usually preferred because it also delivers the shadow price lambda* directly, which the tangency method does not.
When does an inequality constraint actually bind, and why does it matter?
Test the unconstrained optimum first. If the free peak already satisfies g <= c, the constraint is slack, complementary slackness forces lambda = 0, and the answer is just the unconstrained optimum. If the free peak violates g <= c, the constraint binds (g = c) and you solve the ordinary Lagrangian, expecting lambda* >= 0. It matters because assuming a constraint binds when it does not — or vice versa — gives the wrong bundle and loses the shadow-price interpretation.
Can AI help me with constrained optimisation in ECON5005?
Yes, as a study aid. Sia, the AskSia AI tutor, can explain the method step by step: how to set up the Lagrangian, write the first-order conditions, eliminate the multiplier, impose the constraint, and interpret lambda* as a shadow price — using practice problems with your own numbers so you build the reasoning. It does not sit your quizzes, mid-semester or final exam or hand you assessment answers, and it cannot promise a grade. Treat it as a tutor that helps you work each step yourself, and confirm the exact assessment rules, weights and calculator policy on Canvas.
Studying with AI? Sia — free AI economics tutor works through ECON5005 step by step.
Exam move
Constrained optimisation is a method you drill until it is automatic, not a set of facts to memorise. Rehearse one clean pattern: write the Lagrangian L = f + lambda(c - g), take the three first-order conditions, eliminate lambda to get the tangency relation (MRS = price ratio), impose the constraint to pin the bundle, then always interpret lambda* as a shadow price. For inequality constraints, test the unconstrained peak first to decide whether the constraint binds or is slack. Because a full multi-part Lagrangian question anchors the final exam, practise past-style utility- and production-maximisation problems end to end under time (about one minute per mark, so a 12-mark question is roughly 12 minutes), finishing each with the interpretation line rather than stopping at the numbers. Confirm the exact assessment weights, and the open/closed-book and calculator policy, on your Canvas page.
