ECON6002 · Macroeconomic Analysis
One-Period Macroeconomic Models
The first topic of ECON6002 Macroeconomic Analysis at the University of Sydney builds modern macro from its micro-foundations: a representative household maximises utility subject to a budget constraint, a competitive firm maximises profit subject to a production function, and the two plans meet in a single labour market cleared by the real wage W/P. There is no time, no capital accumulation and no uncertainty yet — just one period in which you learn the toolkit reused all semester.
The household's choice between consumption and leisure delivers the labour-supply curve (its marginal rate of substitution equals the real wage); the firm's hiring decision delivers labour demand (the marginal product of labour equals the real wage). Two solution techniques are introduced here and must always agree — the substitution method and the Lagrange method. This block (Topics 1–6) is tested by the closed-book, no-formula-sheet in-semester test, where marks reward derivation and the correct sign of a comparative-static shock, not recall.
What this chapter covers
- 011. Micro-foundations — every aggregate outcome is the solution to an optimisation problem (rational, optimising agents)
- 022. Two solution techniques — substitution (fold the constraint into the objective) vs Lagrange (add it with a multiplier λ); both give the same FOC
- 033. Two-good consumer optimum — the indifference curve is tangent to the budget line, so MRS = the price ratio p₁/p₂ (utility is ordinal only)
- 044. Household consumption–leisure choice — max U(C, L) s.t. PC = W(1−L) + π, giving the FOC MRS between leisure and consumption = W/P
- 055. Labour supply and the wealth effect — non-labour income π (or a stronger taste for leisure) shifts labour supply left; with log utility and π = 0 it is vertical
- 066. Firm production and labour demand — max PY − WN with Y = A·Kᵃ·N¹⁻ᵃ (capital fixed), giving MPL = (1−α)A·Kᵃ·N⁻ᵃ = W/P
- 077. Labour-market equilibrium — supply meets demand at the market-clearing real wage w* and employment N*
- 088. Comparative statics — a positive TFP (A) shock shifts labour demand right, raising both w* and N*; the one-period seed of the RBC model
Firm labour demand and a productivity shock
- +3(a) Real profit is P·A·Kᵃ·N¹⁻ᵃ − W·N. The marginal product of labour is MPL = (1−α)A·Kᵃ·N⁻ᵃ = (2/3)·6·27^(1/3)·N^(−1/3). Since 27^(1/3) = 3, this is MPL = 12·N^(−1/3). Setting MPL = w gives labour demand.
- +2(b) Solve MPL = w: 12·N^(−1/3) = 4 ⇒ N^(1/3) = 3 ⇒ N = 27 workers hired.
- +4(b cont.) Output Y = A·Kᵃ·N¹⁻ᵃ = 6·27^(1/3)·27^(2/3) = 6·3·9 = 162. Check labour's share: real wage bill = w·N = 4·27 = 108 = (1−α)·Y = (2/3)·162, exactly as Cobb–Douglas requires.
- +3(c) A higher A raises MPL at every N, so labour demand shifts right (the firm will pay more for each worker). Along an upward-sloping labour-supply curve, the new equilibrium has both a higher real wage w* and higher employment N* — the one-period version of the RBC co-movement of wages and hours.
Key terms
- Micro-foundations
- The principle that macroeconomic relationships are derived from the optimising behaviour of individual households and firms rather than assumed at the aggregate level. It is the organising idea behind the entire unit.
- Marginal rate of substitution (MRS)
- The slope of an indifference curve — the rate at which a household will trade one good (or leisure) for another while keeping utility constant. At the optimum it equals the relevant relative price.
- Real wage (W/P)
- The nominal wage W divided by the price level P — the amount of consumption goods an hour of work buys. It is the price of leisure and the single price that clears the one-period labour market.
- Marginal product of labour (MPL)
- The extra output from one more unit of labour, (1−α)Y/N under Cobb–Douglas. With capital fixed it diminishes as N rises, and setting MPL = W/P gives the firm's labour-demand curve.
- Labour supply and demand
- Labour supply comes from the household FOC (MRS between leisure and consumption = W/P); labour demand comes from the firm FOC (MPL = W/P). Their intersection is one-period labour-market equilibrium.
- Wealth effect on labour supply
- A rise in non-labour income π makes the household richer without working, so it consumes more leisure (a normal good) and supplies less labour at every wage — labour supply shifts left.
- Substitution vs Lagrange method
- Two equivalent ways to solve a constrained optimisation: fold the constraint into the objective (substitution) or attach it with a multiplier λ (Lagrange). Both must yield the same first-order condition.
- Ordinal utility
- Utility numbers convey only a ranking of bundles, not a cardinal magnitude. Any increasing transformation (e.g. taking logs) represents the same preferences and leaves the MRS and every optimum unchanged.
One-Period Macroeconomic Models FAQ
Can AI help me with one-period macroeconomic models?
Yes — ask Sia to walk through any one-period macroeconomic model problem or concept step by step, the way University of Sydney tests it. Sia is an AI tutor that explains the derivation — setting up the household or firm problem, taking the first-order condition, and signing a comparative-static shock — so you understand the method rather than just seeing an answer.
What is a one-period macroeconomic model?
It is the simplest general-equilibrium model in ECON6002: a single time period with no capital accumulation and no uncertainty. A representative household chooses consumption and leisure, a competitive firm chooses how much labour to hire, and the real wage adjusts until the two plans are consistent. It introduces the optimisation toolkit — objective, constraint, first-order condition — that every later topic reuses.
Why does the MRS equal the real wage at the household's optimum?
Because the household trades leisure for consumption until the rate at which it is willing to give up leisure (the MRS between leisure and consumption) equals the rate at which the market lets it — the real wage W/P, which is the price of an hour of leisure measured in consumption goods. Inverting this condition in employment–wage space traces the labour-supply curve.
What is the difference between the substitution and Lagrange methods?
They are two routes to the same optimum. The substitution method solves the constraint for one variable, folds it into the objective, and takes an ordinary first-order condition — quick when the constraint is easy to rearrange. The Lagrange method attaches the constraint with a multiplier λ (the shadow value of relaxing it) and sets the partial derivatives to zero — it scales to many variables and hands you λ for free. Both must give the same optimality condition, which is a useful self-check.
How does a productivity (TFP) shock affect the labour market?
A rise in total factor productivity A raises the marginal product of labour at every level of employment, so labour demand shifts right. Along an upward-sloping labour-supply curve, the equilibrium real wage and employment both rise. This is the one-period seed of the Real Business Cycle mechanism you meet in Topics 5–6, where a technology shock drives the co-movement of wages and hours.
Which assessment tests the one-period model, and is it closed-book?
The one-period model is Topic 1, part of the Topics 1–6 block examined by the in-semester test — a supervised, in-person, closed-book paper with no formula sheet, made up of extended-response questions. You must be able to set up and solve a constrained optimisation from scratch, so practise deriving the FOCs by hand rather than memorising results. Always confirm the exact weighting and timing on the official unit outline in Canvas.
Studying with AI? Sia — free AI economics tutor works through ECON6002 step by step.
Exam move
Treat the one-period model as the template for the whole unit rather than a stand-alone topic: master writing the objective, imposing the binding constraint, taking the first-order condition, and signing the shock, and you reuse those four moves from here to the New Keynesian model. Drill both solution techniques on the same problem — substitution and Lagrange — and check that they give the identical FOC; that habit both catches algebra slips and earns setup marks. Because the in-semester test is closed-book with no formula sheet, rehearse the household FOC (MRS = W/P) and the firm FOC (MPL = W/P) until you can reproduce them cold, and be able to draw the consumption–leisure diagram and the labour-market diagram with a shock cleanly labelled. Practise the comparative statics explicitly: for each shock (a change in A, K, π, the taste for leisure γ, or the price level P), state which parameter moves, whether a curve shifts or you move along it, and the resulting sign of w* and N*. The classic traps to pre-empt are confusing α with 1−α in the firm FOC, treating utility levels as cardinal, and mixing a movement along labour supply with a shift of it. Ask Sia to generate similar one-period optimisation problems and to check each step of your derivations as you work through Tutorial 1.
