ECON6002 · Macroeconomic Analysis
Real Business Cycle Model, Part 2
ECON6002 Macroeconomic Analysis at the University of Sydney reaches its methodological core in Real Business Cycle Model, Part 2: how you actually solve a DSGE model once it is written down. For a realistic depreciation rate (0 < δ < 1) the Real Business Cycle system has no closed-form solution, so you find the non-stochastic steady state, log-linearise every equilibrium equation with a first-order Taylor approximation, and solve the resulting linear system under the Blanchard–Kahn determinacy conditions — numerically, in Dynare. The dynamics are then read off impulse-response functions to a one-off TFP shock, where investment moves most, output next, and consumption least. On the in-semester test this topic is examined conceptually (what a DSGE solution is, why we linearise, why the result is numerical); for the final it is the foundation for the whole New Keynesian block.
What this chapter covers
- 011. What a DSGE “solution” is — policy functions of the state variables (capital and the shock); why 0 < δ < 1 admits no closed form
- 022. The steady state — the no-shock resting point the whole approximation is taken around
- 033. Log-linearisation — a first-order Taylor approximation written in log-deviations (“hats”) from steady state
- 044. Rule 1 — products and ratios become sums and differences of hats (the production function)
- 055. Rule 2 — an additive rate divides by the steady-state sum (the r + δ trap)
- 066. The linear system A·ᴇₜXₜ₊₁ = BXₜ + Dεₜ and the Blanchard–Kahn determinacy conditions
- 077. Calibration — pinning deep parameters (β, K/Y, I/Y, C/Y) to long-run data ratios
- 088. Impulse-response functions — the dynamics of a +1% TFP shock; the King–Rebelo volatility ranking
- 099. Dynare — the .mod file, the K(−1) time convention, and why the solution is numerical
Calibrating the RBC steady state — the great ratios
- +1(a) The discount factor comes straight from the Euler equation 1 = β(1+r): β = 1/(1+r) = 1/1.015 ≈ 0.985.
- +1(b) Capital demand at the steady state is r + δ = α·Y/K, so K/Y = α/(r + δ) = 0.36/(0.015 + 0.025) = 0.36/0.04 = 9.
- +1In steady state investment just replaces depreciated capital, I = δK, so the investment share is I/Y = δ·(K/Y) = 0.025 × 9 = 0.225.
- +1(c) The resource constraint in shares gives C/Y = 1 − I/Y − G/Y = 1 − 0.225 − 0.20 = 0.575.
- +1Name the pinning equation beside each number: the Euler equation gives β, capital demand gives K/Y, δ·(K/Y) gives I/Y, and C/Y is the resource-constraint residual.
- +1Sanity check: the three uses of output sum to one (0.575 + 0.225 + 0.20 = 1), consumption is the largest use, β < 1, and K/Y = 9 quarters (≈ 2.25 years of output held as capital) is an empirically standard value.
Key terms
- DSGE solution
- A description of every endogenous variable as a function of the state variables (predetermined capital and the current shock) — the “policy functions”. For 0 < δ < 1 these are unknown, generally non-linear functions, so the model has no analytical solution and must be approximated.
- Steady state
- The point the economy settles at with the shock switched off and all variables constant. It is found by dropping time subscripts and expectations from the equilibrium conditions, and it is the point around which the model is linearised.
- Log-linearisation
- A first-order Taylor approximation taken around the steady state, done equation by equation, that turns a non-linear DSGE system into a linear one in log-deviations. Justified because business cycles are small fluctuations around a trend.
- Log deviation (hat)
- The proportional gap of a variable from its steady-state value, x̂ₜ = log Xₜ − log X̄ ≈ (Xₜ − X̄)/X̄. A rate added to a constant is instead handled as a level deviation over the steady-state sum.
- Blanchard–Kahn conditions
- The determinacy requirement for a unique, stable solution: the number of explosive eigenvalues of the linear system must equal the number of forward-looking (jump) variables. Too few gives indeterminacy (sunspots); too many gives an explosive path.
- Impulse-response function (IRF)
- The time path of each variable following a single, unexpected, never-repeated shock, starting from the steady state. For a +1% TFP shock, investment jumps most, output next and consumption least, then all decay back.
- Calibration
- Choosing deep parameters so the model's long-run ratios match observed data targets (capital's share, the real rate, the great ratios), rather than estimating them from a likelihood. Each target pins one parameter through one steady-state equation.
- Dynare
- Software that translates a .mod file into MATLAB to solve and simulate a DSGE model. Its time convention dates a variable at when it is decided, so capital chosen last period enters as K(−1); a missing semicolon is the most common failure.
Real Business Cycle Model, Part 2 FAQ
Can AI help me with the Real Business Cycle model and log-linearisation?
Yes — ask Sia to walk through any Real Business Cycle log-linearisation or DSGE-solution problem or concept step by step, the way University of Sydney tests it. Sia is an AI tutor that explains the method — finding the steady state, applying the two log-linearisation rules, checking Blanchard–Kahn determinacy, reading an impulse response — so you can reproduce it yourself in a closed-book exam. It builds your understanding rather than doing your assessments for you.
Is Topic 6 (RBC Part 2) examined in ECON6002?
Yes, but at different depths. On the 30% in-semester test it is examined only conceptually — what a DSGE solution is, why we log-linearise, and why the result is generally numerical — not Dynare syntax or impulse-response interpretation. For the 55% final it is not directly examined but is foundational: the log-linearised Euler equation becomes the New Keynesian dynamic IS curve, and log-linearisation underlies the whole provided formula sheet.
Do I need to know Dynare code for the ECON6002 exam?
No. The exams are closed-book and do not test Dynare syntax. You should understand conceptually what Dynare does — it takes the steady state, log-linearises, checks the Blanchard–Kahn conditions and simulates — and know its time convention (a variable decided last period is dated at −1, so capital appears as K(−1)). The mechanical work you are tested on is the log-linearisation by hand.
Why can't the Real Business Cycle model be solved analytically?
Because with a realistic depreciation rate (0 < δ < 1) the equilibrium policy functions are unknown, generally non-linear functions of the state variables, so no closed form exists. The clean closed-form solution appears only in the knife-edge special case of full depreciation (δ = 1); otherwise you approximate around the steady state and solve numerically.
How do I log-linearise an equation with an additive term like r + δ?
Use the two rules. Products and ratios become sums and differences of hats (Rule 1). An additive rate cannot be hatted directly — linearise it as the level deviation divided by the steady-state sum, log(rₜ + δ) ≈ log(r + δ) + (rₜ − r)/(r + δ) (Rule 2). Mixing these two rules up is the single most common error; a good check is that the constants cancel when you subtract the steady state.
What is the difference between an impulse and the propagation mechanism?
The impulse is the one-off exogenous shock (for example a +1% TFP innovation). Propagation is how agents' optimal responses — capital accumulation, expectations, consumption smoothing — spread that single shock into a smooth, multi-period path. Confusing the two is a classic exam slip; note that with full depreciation there is almost no propagation, which is why the general 0 < δ < 1 model is the interesting one.
Studying with AI? Sia — free AI economics tutor works through ECON6002 step by step.
Exam move
Treat Real Business Cycle Model, Part 2 as a method to rehearse, not facts to memorise. First be able to say in one line what a DSGE “solution” is (policy functions of the states) and why 0 < δ < 1 forces you to approximate. Then drill the mechanics that carry the marks: find the steady state by dropping shocks and time subscripts, and log-linearise by hand using the two rules — products and ratios become sums of hats, additive rates divide by the steady-state sum — checking every time that the constants cancel. Practise the calibration bookkeeping until β, K/Y, I/Y and C/Y fall out automatically, naming the equation that pins each. Know the Blanchard–Kahn determinacy condition in one line and be able to describe the impulse-response story to a TFP shock (investment > output > consumption, employment up, all decaying). Because the in-semester test is closed-book and conceptual on this topic, rehearse the explanations aloud; because the final leans on it as foundation, make sure the log-linearised Euler equation is second nature before you meet the New Keynesian block. Work Tutorial 6 by hand, and ask Sia to generate similar log-linearisation and calibration problems and to check each step of your derivations.
