ECON1003 · Quantitative Methods In Economics
Multivariable Calculus
Real economic functions depend on several things at once — output on labour and capital, utility on two goods, demand on its own price, income and other prices. Partial differentiation (vary one input, hold the rest fixed) turns each into a marginal quantity, and everything in this chapter is built from that one move. You take partial and cross-partial derivatives, form the total differential to approximate a percentage change, compute MPL/MPK, an MRTS or an MRS, read partial elasticities by sign, and find an unconstrained optimum by setting both partials to zero. It ends with the unit's flagship, highest-mark technique: the Lagrangian for constrained optimisation — utility-max or cost-min subject to a budget or output constraint, including the corner cases, with λ carrying the shadow-price interpretation. This is Final material and worth the most marks in the topic.
What this chapter covers
- 015.1 Partial derivatives and cross-partials
- 025.2 The total differential and percentage-change approximation
- 035.3 Production: MPL, MPK, isoquants and MRTS
- 045.4 Utility: indifference curves and MRS
- 055.5 Unconstrained optimisation
- 065.6 The Lagrangian, complementary slackness and the shadow price
Worked example: utility maximisation with a Lagrangian
- +1Form the Lagrangian. L = xy + λ(80 − 2x − 4y).
- +1First-order conditions. ∂L/∂x = y − 2λ = 0; ∂L/∂y = x − 4λ = 0; ∂L/∂λ = 80 − 2x − 4y = 0.
- +1Combine the first two (the MRS = price-ratio condition). From them y = 2λ and x = 4λ, so x = 2y.
- +1Substitute into the budget. 2(2y) + 4y = 80 ⇒ 8y = 80 ⇒ y = 10, and x = 2y = 20.
- +1Solve for λ. y = 2λ ⇒ λ = 5.
- +1Interpret. λ = 5 is the shadow price: one extra dollar of budget raises maximum utility by about 5 (the marginal utility of income).
Key terms
- Partial derivative
- The derivative of a multivariable function with respect to one variable, treating the others as constants: ∂f/∂x. It is the marginal effect of that one input — MPL is ∂Q/∂L, MPK is ∂Q/∂K.
- Total differential
- The approximate change in z from small changes in both inputs: dz = (∂z/∂x)dx + (∂z/∂y)dy. Each input's marginal effect times its change, added up — the tool for percentage-change approximations.
- MRS / MRTS
- The marginal rate of substitution (consumption) or technical substitution (production): the slope of an indifference curve or isoquant, equal to the ratio of the two marginal quantities. At an optimum it equals the relevant price ratio.
- Lagrangian
- The method for constrained optimisation: L = objective + λ(constraint). Setting the partials to zero yields the first-order conditions; solving them gives the optimum. The unit's flagship technique and worth the most marks.
- Shadow price (λ)
- The value of the Lagrange multiplier at the optimum: the marginal change in the objective from relaxing the constraint by one unit — the marginal utility of income, or the marginal cost of an output target.
Multivariable Calculus FAQ
How do I take a partial derivative?
Differentiate with respect to the chosen variable while treating every other variable as a constant. For z = x²y + 2y + 4: ∂z/∂x = 2xy (the 2y and 4 are constants in x, so they vanish), and ∂z/∂y = x² + 2 (x² is a constant coefficient). For the functions in this course the cross-partials are equal: fₓₓ = fₕₓ.
Why is the Lagrangian worth practising most?
Because it is the unit's flagship technique and carries the most marks in the topic. A full solution sets up L = objective + λ(constraint), writes all the first-order conditions, combines them (this gives MRS = price ratio), substitutes into the constraint to solve for the bundle, and interprets λ as the shadow price. Practise the corner cases (inequality constraints, complementary slackness) too.
What does the multiplier λ actually mean?
At the optimum, λ is the shadow price — how much the objective improves if you relax the constraint by one unit. In a utility-max problem it is the marginal utility of income (an extra dollar of budget); in a cost-min problem it is the marginal cost of producing one more unit of the target output. Always state this interpretation; it is usually a mark.
Exam move
This topic is Final-only and high-value, so invest in it. Get partial derivatives automatic (one variable at a time, the rest constant), then build up: total differential for percentage-change approximations, MPL/MPK and MRS/MRTS as ratios of marginals, partial elasticities by sign. The decisive skill is the Lagrangian — drill the full routine until it is reflex: form L, write every FOC, combine to MRS = price ratio, substitute into the constraint, solve, then interpret λ as the shadow price. Practise the inequality-constraint and complementary-slackness cases, because that is where the hardest marks sit.
