ECON6002 · Macroeconomic Analysis
Real Business Cycle Model, Part 1
ECON6002 Macroeconomic Analysis at the University of Sydney reaches its first fully Dynamic Stochastic General Equilibrium (DSGE) model here: the Real Business Cycle (RBC) model, which takes the Cass–Ramsey growth model and adds stochastic technology shocks and endogenous labour. In the classical tradition, business cycles are the Pareto-optimal response of optimising households and firms to real shocks — there is no market failure and no role for monetary policy. This chapter sets up the firm and household problems, derives the intratemporal labour condition and the (now stochastic) consumption Euler equation, closes general equilibrium, and solves the one analytic special case (δ=1) by guess-and-verify, giving the constant saving rate s = αβ, constant labour, and output as an AR(2) process.
What this chapter covers
- 011. Business cycles & stylised facts — comovement, relative volatility (I ≫ Y > C smoother), persistence
- 022. The RBC/DSGE framework — Cass–Ramsey + stochastic shocks + endogenous labour, rational expectations
- 033. Classical dichotomy — real outcomes independent of nominal variables; no monetary-policy role
- 044. Firm side — Cobb–Douglas production Yₜ = Kₜ^α(AₜLₜ)^(1−α); factor demands wₜ, rₜ
- 055. Shocks — technology & government spending as AR(1) processes; no shocks ⇒ no cycles
- 066. Household FOCs — intratemporal labour condition and the stochastic consumption Euler equation
- 077. General equilibrium — resource constraint + factor clearing; count equations = 8 endogenous variables
- 088. The δ=1 special case — guess-and-verify ⇒ s = αβ, constant L, output as an AR(2) around trend
Guess-and-verify: the constant saving rate s = αβ
- +2(a) The intertemporal optimality condition is the consumption Euler equation 1 = β·Eₜ[(Cₜ/Cₜ₊₁)(1 + rₜ₊₁)]; keep the expectations operator Eₜ because the future return is uncertain.
- +3(b) Guess Kₜ₊₁ = s·Yₜ with s constant, so consumption is Cₜ = Yₜ − Kₜ₊₁ = (1 − s)Yₜ. Substitute this and 1 + rₜ₊₁ = α·Yₜ₊₁/(s·Yₜ) into the Euler equation: 1/[(1−s)Yₜ] = β·Eₜ{[α·Yₜ₊₁/(s·Yₜ)]·1/[(1−s)Yₜ₊₁]}.
- +2The Yₜ₊₁ terms and the (1 − s) factors cancel, and every remaining quantity is known at t, so Eₜ drops out: 1/Yₜ = βα/(s·Yₜ).
- +2Multiply through by Yₜ: 1 = βα/s, hence s = αβ — a constant, independent of the shock — which verifies the guess. So Kₜ₊₁ = αβ·Yₜ and Cₜ = (1 − αβ)Yₜ.
- +1(c) With α = 0.36 and β = 0.96, s = αβ = 0.3456, so the household saves about 34.6% of output and consumes Cₜ = 0.6544·Yₜ.
Key terms
- DSGE model
- Dynamic Stochastic General Equilibrium model — the workhorse framework (Real Business Cycle and later New Keynesian) in which optimising agents make dynamic choices under uncertainty and rational expectations, and markets clear in general equilibrium.
- Real Business Cycle (RBC) model
- A DSGE model in the classical tradition where business cycles are the optimal, Pareto-efficient response of households and firms to real shocks — chiefly temporary technology (TFP) shocks. It is the Cass–Ramsey growth model plus stochastic shocks and endogenous labour.
- Rational expectations
- Model-consistent, statistically correct expectations: agents know the model structure and the distributions of the shocks, so Eₜ is the conditional expectation given information at time t. For any variable already known at t, Eₜ Xₜ = Xₜ.
- Classical dichotomy / money neutrality
- The property that real outcomes are independent of nominal variables. In the RBC world nominal shocks do not move real quantities, so there is no role for monetary policy in smoothing cycles — the point of departure for the New Keynesian model.
- TFP / technology shock
- A stochastic innovation to total factor productivity A, modelled as an AR(1) process in logs. It is the main impulse driving fluctuations in the RBC model: no shocks means no cycles.
- Consumption Euler equation
- The intertemporal optimality condition 1 = β·Eₜ[(Cₜ/Cₜ₊₁)(1 + rₜ₊₁)]: at the optimum the household is indifferent between consuming a dollar today and saving it at the return rₜ₊₁. Now stochastic, so it sits inside the expectations operator Eₜ.
- Intratemporal labour condition
- The within-period optimality b/(1 − Lₜ) = wₜ/Cₜ: the marginal disutility of working equals the marginal utility of the consumption the extra wage can buy. It links this period's hours to this period's wage and consumption.
- Guess-and-verify
- A solution method: propose a functional form for the endogenous variables (here a constant saving rate), substitute it into the equilibrium equations, and confirm it satisfies them. It yields the RBC model's only closed-form solution, under δ = 1. The state variables Kₜ and Aₜ are predetermined, known before period-t decisions.
Real Business Cycle Model, Part 1 FAQ
Can AI help me with the Real Business Cycle (RBC) model?
Yes — ask Sia to walk through any Real Business Cycle model problem or concept step by step, the way University of Sydney tests it. Sia is an AI tutor that explains: give it a household problem, an Euler-equation derivation, or the δ=1 guess-and-verify, and it talks you through the mechanism and the algebra rather than just handing over a final answer, so you can reproduce the derivation yourself in the closed-book test.
What is the Real Business Cycle model in one sentence?
It is the first fully micro-founded DSGE model in ECON6002: a Cass–Ramsey growth model with stochastic technology shocks and endogenous labour, in which business cycles emerge as the optimal, Pareto-efficient response of rational households and firms to real shocks. Because it lives in the classical dichotomy, nominal variables are neutral and there is no role for monetary policy.
Why is the saving rate s = αβ only a special case?
The clean closed form s = αβ requires full depreciation (δ = 1), zero government spending and log utility — the Long–Plosser case where all capital is used up each period, so Kₜ₊₁ = Yₜ − Cₜ. With realistic depreciation the model is non-linear and has no pencil-and-paper solution; you log-linearise around the steady state and solve numerically (that is Part 2). A common exam error is presenting s = αβ as the general RBC result.
Why does the consumption Euler equation sit inside an expectations operator here?
Because the future gross return rₜ₊₁ is uncertain — it depends on next period's technology and capital, which are not yet known at t. So the optimality condition is 1 = β·Eₜ[(Cₜ/Cₜ₊₁)(1 + rₜ₊₁)], and you must carry Eₜ until the randomness genuinely cancels (as it does in the δ=1 special case). Dropping the operator too early is one of the most common ways students lose marks.
Is the RBC model examined in ECON6002, and where?
Yes. Topic 5 (RBC Part 1) is examined on the In-Semester Test (the midterm) — a supervised, closed-book paper with NO formula sheet, made of extended-response questions covering Topics 1–6. RBC does not appear on the final exam (which covers Topics 7–10). The practical implication is that you must be able to reproduce the household FOCs, the Euler equation and the guess-and-verify saving rate entirely from memory.
How should I revise the RBC model for the exam?
Drill the derivations until you can reproduce them without notes: the household problem, the intratemporal labour FOC, the stochastic Euler equation, the equation-counting check (8 equations = 8 endogenous variables), and the δ=1 guess-and-verify to s = αβ and constant labour. Practise by hand from the tutorial questions during the revision/SWOTVAC period, and be ready to explain why log utility pins labour and why δ=1 gives little propagation. You can also ask Sia to generate RBC-style practice and check each step of your working.
Studying with AI? Sia — free AI economics tutor works through ECON6002 step by step.
Exam move
Treat RBC Part 1 as the payoff of everything before it: the same constrained-optimisation and Euler-equation machinery from the two-period and Cass–Ramsey models returns, now with an expectations operator and endogenous labour. Because Topic 5 is examined on the closed-book, no-formula-sheet In-Semester Test, the single most valuable habit is rehearsing derivations until you can reproduce them cold — set up the household problem, derive the intratemporal labour condition and the stochastic consumption Euler equation, count that the eight equilibrium equations match the eight endogenous variables, then run the δ=1 guess-and-verify to s = αβ (and, if asked, the constant labour L and the AR(2) for output). Learn the boundaries the markers probe: why log utility makes labour constant (income and substitution effects cancel), why δ=1 delivers only weak propagation, and why the classical dichotomy leaves no role for monetary policy. Work the tutorial problems by hand during the revision/SWOTVAC period rather than reading solutions, and ask Sia, an AI tutor that explains step by step, to check your working and generate fresh RBC-style practice in the way University of Sydney tests it.
