Real Business Cycle (simplified DSGE)
How a stochastic productivity shock propagates through a small dynamic equilibrium economy.
Built and reviewed by Stephen Omukoko Okoth
Mathematical Economist · ex-Morgan Stanley FI · Equilar
Theory
What the model says, and why
Modern macroeconomics is dominated by Dynamic Stochastic General Equilibrium (DSGE) models. They’re the analytical engine behind central banks (Federal Reserve’s SIGMA, the ECB’s NAWM, Norges Bank’s NEMO), academic macro, and most serious modern policy analysis. The starting point — and still the cleanest teaching version — is the Real Business Cycle model of Kydland & Prescott (1982).
The setup is striking in its sparseness. A single representative household maximizes expected lifetime utility subject to a budget constraint. A representative firm produces output via a Cobb-Douglas technology. The only source of randomness is an AR(1) process on total factor productivity:
s.t. Cₜ + Iₜ = Yₜ = Aₜ · Kₜ^α · Lₜ^(1−α)
Kₜ₊₁ = (1−δ)·Kₜ + Iₜ
ln Aₜ₊₁ = ρ · ln Aₜ + εₜ₊₁
The deep claim of RBC. Business cycles aren’t market failures. They’re the optimal response of a perfectly competitive economy to productivity shocks. When productivity rises, output rises mechanically; people consume more, but they also save more because the marginal product of capital is high — investment booms, capital accumulates. Productivity reverts; output falls; investment collapses faster than consumption. The cycle is endogenous, equilibrium, and Pareto-optimal.
Solution method. The model has no closed form. It’s solved by log-linearizing the first-order conditions around a non-stochastic steady state and tracing impulse responses. The playground below uses a heavily simplified version of this approach — the qualitative shape of the responses is right even though the quantitative magnitudes would require a proper Dynare-style solution.
Where it succeeds. RBC matches the volatility of investment relative to output and the persistence of recessions. It’s the foundation on which New Keynesian DSGE models add nominal rigidities (sticky prices, wages), monetary policy rules, and richer shock structures.
Where it fails. It can’t produce monetary non-neutrality — by construction, money plays no role. It can’t explain unemployment (the labor market always clears). It struggles with financial crises. New Keynesian DSGE addresses these by adding frictions, but the RBC core remains.
Interactive playground
Move the parameters, watch the equilibrium move
Parameters
Calibration
Steady state
r* (real rate)
3.51%
1/β − 1 + δ
K/Y ratio
9.40
Capital-output
C/Y
76%
Consumption share
I/Y
24%
Investment share
Impulse responses
One-time +1σ productivity shock at t = 0
Standard RBC shapes: productivity (green dashed) follows AR(1) decay; output rises and decays with it; consumption is smooth (consumption smoothing); investment is volatile, rising sharply then collapsing as the shock fades; capital builds up slowly. The investment volatility relative to output is one of the model’s most-celebrated quantitative successes.
In the classroom
How to teach it well
Two-lecture introduction. First lecture: the household’s problem (Euler equation, intertemporal substitution), the firm’s problem (rental rates), and how the two combine. Second lecture: log-linearization around the steady state, the matrix algebra of impulse responses, and a hands-on calibration exercise using the playground above.
The key intuition to drill. Why does investment respond so much more than consumption? Because the productivity shock raises the marginal product of capital, which raises the return to investing. Households substitute intertemporally — saving more now to consume more later. Consumption is smoothed; investment carries the volatility. This single mechanism is the core of the model.
Common student frustration. “The model has no government, no money, no banks, no unemployment — what good is it?” The honest answer: it’s the chassis. Once students understand how the chassis behaves under productivity shocks, you can add government (fiscal RBC), monetary policy (NK-DSGE with sticky prices), unemployment (search-and-matching), banks (financial frictions). Each extension is a separate paper.
Where to go from here. Clarida-Gali-Gertler (1999) for NK monetary policy, Smets-Wouters (2003, 2007) for the canonical estimated NK-DSGE, Christiano-Eichenbaum-Evans (2005) for richer sticky-price mechanics, Gertler-Karadi (2011) for unconventional policy. The DSGE literature is vast; the RBC model is its first stop.
Caveat for this playground. The impulse responses shown use a simplified linear approximation, not a full Dynare-style solution. The qualitative shapes are correct (consumption smooth, investment volatile, capital slow); precise magnitudes would differ. For research-grade work, students should graduate to Dynare or a Python equivalent (econ_ark, dolo).