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How does a single technology shock propagate through capital accumulation and labor choice in a minimal dynamic framework without government or nominal frictions?
The simplified RBC model strips the full Kydland-Prescott (1982) framework down to its bare essentials: one representative household, one production technology, one exogenous shock, and no government sector. This pedagogical version appears in Romer (2019, Ch. 5) and Williamson (2018, Ch. 11) as the entry point to dynamic stochastic general equilibrium modeling. The goal is to isolate the core propagation mechanism -- technology shocks hitting an economy with capital accumulation and endogenous labor supply -- before layering on fiscal shocks, nominal rigidities, or financial frictions.
The mechanism is straightforward. A positive technology shock raises the marginal product of both capital and labor. Households respond along two margins: they work more (intratemporal substitution toward labor, which is temporarily more productive) and they save more (intertemporal substitution toward future consumption, since the return to capital is temporarily high). Capital accumulation carries the shock forward, generating persistence in output even after the productivity innovation has died out. This is the entire propagation story -- no multiplier-accelerator, no sticky prices, no financial amplification.
This simplified version is the standard first DSGE model in graduate macro sequences at most research universities. It is used as a building block: once students can solve the steady state, derive the Euler equation, log-linearize, and compute impulse responses on this model, they add government spending (full RBC), then sticky prices (New Keynesian), then heterogeneous agents (HANK). The stripping-down is the point -- every equation present is load-bearing.
Three blocks: a representative household choosing consumption, labor, and saving; a representative firm operating Cobb-Douglas technology with stochastic TFP; and a resource constraint tying output to consumption and investment. No government, no transfers, no taxes. The household owns the capital stock and rents it to the firm.
Equilibrium is pinned by two optimality conditions (the Euler equation and the labor-leisure condition), one production function, one capital accumulation law, and one AR(1) technology process. That gives five equations in five unknowns per period: output, consumption, investment, hours, and the capital stock. Factor prices (wage and rental rate) are determined residually from the firm's marginal product conditions.
Solution proceeds by finding the deterministic steady state, log-linearizing around it, and solving the resulting linear rational expectations system. The Blanchard-Kahn condition requires exactly one unstable eigenvalue (consumption is the jump variable) and two stable roots (capital and technology are the states). The solved model yields policy functions mapping states to controls and a state transition law that generates impulse responses.
Graduate macro courses at institutions like MIT, Chicago, and Minnesota use this model as the first complete DSGE students solve. It teaches steady-state computation, the Euler equation, log-linearization, Blanchard-Kahn conditions, and impulse response analysis in one compact package. Romer (2019) and Williamson (2018) build entire chapters around it.
The simplified RBC serves as the baseline when researchers add one friction at a time -- sticky prices, habit formation, investment adjustment costs -- to measure the marginal contribution of each friction. Starting from a stripped-down version makes the comparison clean because there is no second shock to confuse the attribution.
This model should not be used for policy analysis involving monetary policy, fiscal multipliers, financial crises, or distributional questions. It has no nominal variables, no government, and no heterogeneity. If any of those matter for the question, the right move is a fuller model that nests this one as a special case.
Rate at which the household discounts future utility; pins the steady-state real interest rate via 1/beta - 1.
Output elasticity of capital in the Cobb-Douglas production function; typically calibrated to 0.33--0.36 from U.S. national accounts.
Fraction of the capital stock lost each period; set to match the investment-output ratio in steady state.
Governs labor supply elasticity. Higher values make hours less responsive to wage changes.
AR(1) coefficient on log TFP. Controls how long a productivity shock reverberates through the economy.
The sole exogenous driving process. Follows a stationary AR(1) in logs with Gaussian innovations.
Endogenous state variable linking periods through the accumulation equation K_{t+1} = (1 - delta)K_t + I_t.
A single household with perfect access to capital markets stands in for the entire economy. No idiosyncratic risk, no borrowing constraints.
If violated: Wealth heterogeneity generates different marginal propensities to consume, requiring TANK or HANK extensions.
All factor prices adjust instantly each period to equate marginal products to factor returns.
If violated: Sticky prices create a role for monetary policy and demand-driven fluctuations -- the New Keynesian channel.
Y_t = A_t K_t^{\alpha} N_t^{1-\alpha} with constant returns to scale and unitary substitution elasticity.
If violated: CES production alters how factor income shares respond to shocks, changing the consumption-investment split.
The household knows the model and forms expectations using the true conditional distribution of future shocks.
If violated: Bounded rationality or learning changes the speed and shape of shock propagation.
All business cycle variation comes from innovations to A_t. No demand shocks, no fiscal shocks, no financial shocks.
If violated: If demand or financial shocks matter empirically, this model cannot match the data and understates output volatility.
Hours worked are always strictly between zero and the time endowment, so the intratemporal condition holds with equality.
If violated: Corner solutions (unemployment, zero hours) require search-matching or indivisible labor extensions.