Real Business Cycle

Definition

Real business cycle (RBC) theory is a class of dynamic general equilibrium models in which business cycle fluctuations are optimal responses by households and firms to exogenous shocks to total factor productivity (TFP). In the prototypical one-sector model, the sole driving force is a random walk in TFP; monetary variables are neutral in the long run and largely irrelevant for real fluctuations. Output gaps — deviations of output from potential — are not inefficient; they are the efficient response to changed economic circumstances.

Key Ideas

Technology shocks as the driving force. A positive TFP shock raises the marginal product of capital and labour, inducing substitution into work (intertemporal labour substitution) and generating co-movements in output, consumption, and investment that qualitatively mimic business cycle data.
Balanced growth. Under balanced growth, per capita output, consumption, and investment all grow at the same long-run rate $\mu_\lambda / \theta$ (where $\mu_\lambda$ is mean TFP growth and $\theta$ is labour's share). The consumption–output and investment–output ratios ("great ratios") are therefore stationary, implying that the three series are cointegrated — a testable prediction.
A single common stochastic trend. The balanced-growth implication is that log output, consumption, and investment share a single I(1) common stochastic trend (TFP itself); see Common Stochastic Trends.
Money neutrality. In most RBC models, nominal variables play no causal role in real fluctuations; monetary policy is superneutral.
No exploitation of Phillips tradeoff. Because output fluctuations are efficient and the economy always operates on its production possibilities frontier (conditional on the shock realisation), there is no scope for aggregate demand management.

How It Works

Canonical Structure

Output is produced via a constant-returns Cobb-Douglas function:

$Y_t = \lambda_t K_t^{1-\theta} N_t^\theta$

where $\lambda_t$ is TFP, $K_t$ capital, $N_t$ labour. TFP follows a logarithmic random walk:

$\log(\lambda_t) = \mu_\lambda + \log(\lambda_{t-1}) + \xi_t, \qquad \xi_t \overset{iid}{\sim} (0, \sigma^2)$

The representative household maximises discounted expected utility subject to the resource constraint $Y_t = C_t + I_t$ and capital accumulation $K_{t+1} = (1-\delta)K_t + I_t$ . On the balanced-growth path, $C_t$ , $I_t$ , and $Y_t$ all grow at rate $e^{\mu_\lambda / \theta}$ ; $C_t/Y_t$ and $I_t/Y_t$ are constant. With stochastic TFP, these ratios fluctuate around their means, and the variables share the common stochastic trend $\log(\lambda_t)/\theta$ .

Cointegration Implication

Because each of $y_t = \log Y_t$ , $c_t = \log C_t$ , and $i_t = \log I_t$ is I(1) (driven by the random walk in $\log \lambda_t$ ), but $c_t - y_t$ and $i_t - y_t$ are I(0) (they fluctuate around constant means), the system $(y_t, c_t, i_t)$ has cointegration rank 2. This means a cointegrated vector autoregression (VAR), or vector error correction model (VECM), nests the log-linear approximation to any one-sector RBC model. King-Plosser-Stock-Watson (KPSW, 1991) confirmed this cointegration prediction for postwar U.S. data (Augmented Dickey-Fuller (ADF) test on $c - y$ : $-4.21$ ; on $i - y$ : $-3.99$ ; both significant at 1%).

Extensions

Time-to-build (Kydland-Prescott 1982): Multi-period investment technology that generates hump-shaped investment responses; each stage of a multi-period project inherits the common TFP trend.
Multi-sector models (Long-Plosser 1983): Multiple sectors each with sector-specific productivity trends; can generate multiple common stochastic trends.
Endogenous growth (King-Rebelo 1988): Permanent changes in tax rates or knowledge accumulation affect growth rates, producing permanent changes in the level of all series — still cointegrated.
Nominal extensions: Adding money via a cash-in-advance constraint or money-in-utility generates additional cointegrating relations (money demand, Fisher parity) without disturbing the balanced-growth restrictions on real variables.

Why It Matters

RBC models provided the first internally consistent, model-based explanation for why output, consumption, and investment comove over the business cycle — and the first framework with sharp econometric predictions testable without ad hoc identifying assumptions. King-Plosser-Stock-Watson (1991) showed how to test these predictions using cointegration techniques: the balanced-growth hypothesis passes its cointegration test, lending credibility to the RBC framework as a long-run description of the U.S. economy. However, variance decompositions reveal that the single balanced-growth shock explains less than half of output variability at business-cycle horizons in systems including nominal variables, pointing toward the need for richer multi-shock models.

Open Questions

The role of a persistent real-interest-rate permanent component in KPSW (1991) has no natural interpretation in standard one-sector RBC models; it may signal credit or financial channels that RBC abstracts from.
Medium-scale Dynamic Stochastic General Equilibrium (DSGE) models (Smets-Wouters 2003) incorporate RBC as the supply side while adding nominal rigidities; whether these retain the balanced-growth cointegration structure at the estimated parameter values is not always verified.
Multi-sector generalisations can have multiple common stochastic trends, making the cointegration-rank prediction endogenous to the model specification.