ECON5005 · Quantitative Tools for Economics
Partial Derivatives & Differentials
This chapter is where single-variable calculus grows a second variable in ECON5005 Quantitative Tools for Economics at the University of Sydney. You learn to differentiate a two-input function one variable at a time (the partial derivative), to combine the separate effects into one linear approximation (the total differential), and to read a function through its level curves — the indifference curves, isoquants and isocost lines that run through the rest of your economics degree. It sits in Week 6, bridging Week 5 (single-variable optimisation) and Week 7 (constrained optimisation and the Lagrangian), and feeds the tangency condition that solves the consumer's problem.
What this chapter covers
- 01Take a partial derivative by holding the other variable constant, reusing the power, product, quotient and chain rules
- 02Read partials as marginal quantities: marginal products MP_K, MP_L and marginal utilities MU_x, MU_y
- 03Write the total differential df = (∂f/∂x₁)dx₁ + (∂f/∂x₂)dx₂ as a first-order approximation to a change
- 04Define a level curve f(x₁,x₂) = c and derive its slope by setting the total differential to zero
- 05Get the slope as minus the ratio of partials: dx₂/dx₁ = −(∂f/∂x₁)/(∂f/∂x₂) — keep the minus sign
- 06Read the same formula three ways: indifference curve (MRS), isoquant (MRTS), isocost (−w/r)
- 07Compute the MRS = MU_x/MU_y and interpret it as goods a consumer will trade at equal utility
- 08Compute the MRTS = MP_L/MP_K and interpret it as inputs a firm can swap at constant output
- 09Classify returns to scale from the degree of homogeneity n in F(tK,tL) = tⁿ F(K,L)
- 10Tell substitution (along one curve) apart from scaling (between curves)
Marginal products, MRTS and returns to scale for a Cobb–Douglas firm
- +1Marginal product of labour — differentiate in L, holding K constant: MP_L = ∂Q/∂L = 4·K^(1/2)·(1/2)·L^(−1/2) = 2·K^(1/2)·L^(−1/2).
- +1Marginal product of capital — differentiate in K, holding L constant: MP_K = ∂Q/∂K = 4·(1/2)·K^(−1/2)·L^(1/2) = 2·K^(−1/2)·L^(1/2).
- +2MRTS is the ratio of marginal products: MRTS = MP_L/MP_K = [2·K^(1/2)·L^(−1/2)] / [2·K^(−1/2)·L^(1/2)] = K^(1/2+1/2)·L^(−1/2−1/2) = K/L.
- +1Evaluate at (K, L) = (9, 4): MRTS = K/L = 9/4 = 2.25. Locally, cutting labour by one unit needs about 2.25 extra units of capital to hold output constant; the isoquant slope is dK/dL = −2.25.
- +1Returns to scale — scale both inputs by t: F(tK, tL) = 4·(tK)^(1/2)·(tL)^(1/2) = t^(1/2+1/2)·4·K^(1/2)·L^(1/2) = t·F(K, L). The degree is n = 1, so the technology has constant returns to scale.
Key terms
- Partial derivative
- The derivative of a multi-variable function with respect to one variable, treating every other variable as a constant; written ∂f/∂x or f_x. All the ordinary rules (power, product, quotient, chain) still apply.
- Total differential
- For y = f(x₁, x₂), dy = (∂f/∂x₁)dx₁ + (∂f/∂x₂)dx₂: the sum of each partial times its variable's change. It is a first-order (linear) approximation to the change in y and is exact only in the limit of infinitesimal moves.
- Level curve
- The set of input pairs {(x₁, x₂): f(x₁, x₂) = c} that hold the function at a fixed value c — a contour line on the function's map. Its slope, dx₂/dx₁ = −(∂f/∂x₁)/(∂f/∂x₂), comes from setting the total differential to zero.
- Indifference curve
- A level curve of a utility function, U(x, y) = c: bundles that give the consumer equal satisfaction. Its slope is −MRS, and for standard preferences it is convex to the origin.
- Isoquant
- A level curve of a production function, Q(K, L) = c: input combinations that yield the same output. Its slope is −MRTS, the absolute value of the marginal-product ratio.
- MRS (marginal rate of substitution)
- MRS = MU_x/MU_y, the amount of good y a consumer will give up for one more unit of good x while staying equally happy; it equals the absolute slope of the indifference curve.
- MRTS (marginal rate of technical substitution)
- MRTS = MP_L/MP_K, the rate at which capital can replace labour while holding output constant; it equals the absolute slope of the isoquant and falls as you move along a convex isoquant.
- Returns to scale
- How output responds when every input is scaled by the same factor t. If F(tK, tL) = tⁿ·F(K, L) the function is homogeneous of degree n: n = 1 is constant returns, n > 1 increasing, n < 1 decreasing. For Cobb–Douglas Q = A·K^a·L^b, n = a + b.
Partial Derivatives & Differentials FAQ
Is this topic examined in ECON5005, and how does it appear?
Yes. Level curves, the total differential, the MRS/MRTS and returns to scale are Week 6 material and usually appear inside a longer multi-part production or utility question rather than as a stand-alone item — for example one part for the partial derivatives, one for the isoquant slope or total differential, and one naming the returns to scale. The topic also underpins Week 7, because the tangency condition that solves the consumer's problem (set the MRS equal to the price ratio) is built from the level-curve slope you learn here. The final is a 2-hour in-person written exam in the formal exam period; check your exact date, and whether it is open- or closed-book, on Canvas.
What is the difference between the MRS/MRTS and returns to scale?
They answer different questions. The MRS and MRTS are slopes along a single curve: they measure how you can substitute one input for another while holding utility or output constant (moving along one indifference curve or isoquant). Returns to scale asks what happens when you scale every input by the same factor t at once — moving between curves with the input mix fixed. A question can ask for both, so read carefully which it wants: substitution is a ratio of marginals, scale is the degree of homogeneity n in F(tK, tL) = tⁿ F(K, L).
Can AI help me with partial derivatives and level curves in ECON5005?
Yes, as a study aid. Sia, the AskSia AI tutor, can explain how to take a partial derivative step by step, walk through a similar isoquant-slope or total-differential problem so you can see the method, check where a sign or an exponent went wrong, and quiz you on the difference between MRS, MRTS and returns to scale. It is there to help you understand and practise the technique — not to hand in answers for you. Build the skill with it, then complete the actual assessed quizzes, test and exam yourself, in line with the University of Sydney's academic-integrity rules.
Studying with AI? Sia — free AI economics tutor works through ECON5005 step by step.
Exam move
Anchor everything to one move: set the total differential to zero and solve for the slope. Practise it until dx₂/dx₁ = −(∂f/∂x₁)/(∂f/∂x₂) is automatic, then recognise its three faces — indifference curve (−MRS), isoquant (−MRTS) and isocost (−w/r). Drill partial derivatives on Cobb–Douglas functions daily, always finishing the number at a named point rather than stopping at the symbolic ratio, and keep the minus sign on the slope and the correct variable on top of the ratio. Separately, practise reading returns to scale straight off the exponents (n = a + b for Cobb–Douglas) and confirming it by the tⁿ scaling test, so you never confuse substitution with scale. Because long questions are graded on method, write each partial, then the ratio, then the evaluated value, then a one-line economic reading — partial marks reward every visible step.
