Dynamic Stochastic General Equilibrium

Brock-Mirman and the Prescott Real Business Cycle Model

Alan Lujan

Johns Hopkins University

April 21, 2026

From Consumption to Macroeconomic Fluctuations

The DSGE Agenda

For eleven modules we have studied a single agent facing a stochastic income stream. The intertemporal Euler equation, the precautionary saving motive, and the permanent-income logic all live at the household level.

A working macroeconomic model must close the loop:

Households solve a dynamic optimization problem, taking prices as given
Firms maximize profit, taking prices as given
Markets clear, and prices adjust

The Dynamic Stochastic General Equilibrium (DSGE) program builds aggregate fluctuations from these three ingredients.

Why These Two Models

This lecture studies the two canonical entry points:

Model	Virtue	Cost
Brock-Mirman (1972)	Closed-form consumption rule	Requires 100% depreciation and log utility
Prescott RBC (1986)	Matches output volatility	Requires implausibly elastic labor supply

Brock-Mirman shows how clean the answer can be when assumptions align. Prescott shows how hard it is to match the data once those assumptions relax.

Roadmap

The lecture develops four ideas.

Analytical DSGE is possible only under special assumptions (Brock-Mirman)
General DSGE is numerical: adding depreciation below 100% forces simulation
Labor-leisure choice is the RBC framework’s main addition to the growth model
Labor-volatility puzzle exposes the model’s empirical weakness

The common thread is the consumption Euler equation, now embedded in general equilibrium.

The Brock-Mirman Stochastic Growth Model

The Crucial Assumption: 100% Depreciation

Notice that \(K_{t+1} = Y_t - C_t\), not \((1-\delta) K_t + Y_t - C_t\). Capital depreciates completely within one period.

This is unrealistic on its face. Why assume it?

It collapses the two state variables \((K_t, A_t)\) into a single state \(Y_t\)
The household only cares about income today, not the mix of capital and productivity that produced it
Combined with log utility, it delivers a linear consumption function

Every piece of the closed-form solution depends on this assumption.

The Bellman Equation

Writing the problem recursively in income \(Y_t\):

\[V_t(Y_t) = \max_{C_t} \; \log C_t + \beta \, \mathbb{E}_t \left[ V_{t+1}(Y_{t+1}) \right]\]

The envelope and first-order conditions together yield the Euler equation.

The Euler Equation

Let \(R^k_{t+1} \equiv \alpha A_{t+1} K_{t+1}^{\alpha - 1}\) denote the gross marginal product of capital. The FOC is:

\[\boxed{\; 1 = \beta \, \mathbb{E}_t \left[ R^k_{t+1} \cdot \frac{C_t}{C_{t+1}} \right] \;}\]

This is the standard consumption Euler equation with \(R^k\) playing the role of the interest factor. The special structure of Brock-Mirman comes not from the Euler equation, but from the budget constraint \(K_{t+1} = Y_t - C_t\).

The Proposed Consumption Rule

Conjecture a linear consumption function:

\[C_t = \kappa \, Y_t \qquad \text{where } \kappa = 1 - \alpha \beta\]

Under this rule, \(K_{t+1} = (1 - \kappa) \, Y_t = \alpha \beta \, Y_t\). The marginal propensity to consume is constant and depends only on the two structural parameters.

Does this rule satisfy the Euler equation?

Verifying the Rule: Setup

The MPC is the simple expression \(\kappa = 1 - \alpha \beta\). Intuitively, the household saves exactly the fraction \(\alpha \beta\): the capital share (which governs how much tomorrow’s income rises with saving) times the discount factor (which governs patience).

Verifying the Rule: Euler Check

The Euler equation says \(1 = \beta \, \mathbb{E}_t[\alpha A_{t+1} K_{t+1}^{\alpha-1} \cdot C_t / C_{t+1}]\). Substituting the conjectured rule:

The expectation operator drops out: the \(A_{t+1}\) terms cancel exactly, leaving \(1\). The proposed rule satisfies the Euler equation for every realization, not only in expectation.

Brock-Mirman Dynamics and Steady State

Log-Linear Capital Dynamics

Substituting \(K_{t+1} = \alpha \beta Y_t\) into \(Y_t = A_t K_t^\alpha\) gives:

\[K_{t+1} = \alpha \beta \, A_t \, K_t^\alpha\]

Taking logs, and writing lowercase letters for log variables (\(k_t \equiv \log K_t\), \(a_t \equiv \log A_t\)):

\[\boxed{\; k_{t+1} = \log(\alpha \beta) + a_t + \alpha \, k_t \;}\]

Log capital is an AR(1) process with autoregressive coefficient equal to capital’s share \(\alpha\), driven by the productivity shock \(a_t\).

Log-Linear Output Dynamics

Output in logs is \(y_t = a_t + \alpha k_t\). Combining with the law of motion for \(k\):

\[y_{t+1} = a_{t+1} + \alpha \, k_{t+1} = a_{t+1} + \alpha \log(\alpha \beta) + \alpha a_t + \alpha^2 k_t\]

Output inherits the same autoregressive coefficient as capital. Both variables respond to productivity shocks with serial correlation that depends only on \(\alpha\).

Two Productivity Processes

Brock-Mirman remains tractable under any specification of \(a_t\). Two natural cases:

Random walk: \(a_{t+1} = a_t + \varepsilon_{t+1}\). Shocks are permanent.
White noise: \(a_{t+1} = \bar{a} + \varepsilon_{t+1}\). Shocks are transitory.

These generate very different business-cycle dynamics. The capital share \(\alpha\) controls propagation; the shock process controls the source.

Impulse Response Function

Starting from the steady state, consider a one-period positive productivity shock \(\varepsilon_{t^*} = 1\). Your task below is to fill in the law of motion and generate the paths.

Notice that output under the random walk does not jump to its new level in one period. Capital must accumulate first, and the AR(1) coefficient \(\alpha\) slows that adjustment.

Stochastic Steady State

A nonstochastic economy converges to \(\bar{k}\) where \(\dot{k} = 0\). What is the analogue when shocks are permanent?

Three conventions in the literature:

No-shocks limit: the value to which the economy drifts when \(\varepsilon_t = 0\) for all future \(t\), even though agents expect shocks
Ergodic mean: the mean of \(K_t\) in the stationary distribution
Expected-self point: the state \(\bar{K}\) satisfying \(\mathbb{E}_t[K_{t+1}] = K_t\) when \(K_t = \bar{K}\)

Brock-Mirman is special: all three definitions coincide.

Stochastic Steady State: Derivation

Setting \(K_{t+1} = K_t = \bar{K}\) in the Euler equation and using the exact cancellation of \(A_{t+1}\) terms:

\[1 = \beta \alpha \, A_t \, \bar{K}^{\alpha - 1} \quad \Longrightarrow \quad \bar{k} = \frac{\log(\alpha \beta) + a_t}{1 - \alpha}\]

The steady state depends on the current realization of \(a_t\) but is otherwise the same as in the deterministic version of the model. This coincidence relies on the linearity of the consumption rule.

Why Brock-Mirman Is Special

The closed-form solution rests on three assumptions:

Log utility (CRRA coefficient \(\rho = 1\)): makes income effects and substitution effects cancel
100% depreciation: collapses the state space from \((K, A)\) to \(Y\)
Cobb-Douglas production: gives constant returns to capital in the right sense

Drop any one and the analytical solution breaks. The model is a benchmark, not a realistic description of capital dynamics.

The Prescott Real Business Cycle Model

The Representative Household

Prescott (1986) introduces the canonical RBC setup. The household maximizes:

\[\mathbb{E}_0 \left[ \sum_{t=0}^\infty \beta^t \, u(c_t, \ell_t) \right]\]

subject to the time budget \(n_t + \ell_t = 1\), where \(n_t\) is labor hours and \(\ell_t\) is leisure. The endowment of time in each period is normalized to one.

The household now makes two choices each period: how much to consume, and how much to work.

Utility over Consumption and Leisure

Prescott’s utility function:

\[u(c, \ell) = \frac{(c^{1-\zeta} \, \ell^\zeta)^{1-\rho}}{1-\rho}\]

This is a CRRA transformation of a Cobb-Douglas aggregate of \(c\) and \(\ell\). The weight \(\zeta\) governs the leisure share in the aggregator.

With \(\rho = 1\) (log utility), it collapses to:

\[u(c, \ell) = (1-\zeta) \log c + \zeta \log \ell\]

Why Cobb-Douglas Preferences

Over the past century in the United States, wages have risen fivefold while hours worked per worker have barely moved (Ramey and Francis, 2009).

What utility function keeps the budget share of a good constant when its price changes sharply? Cobb-Douglas.

Any preference class with intertemporal substitution elasticity different from unity would predict a strong trend in hours. The lack of such a trend pins down Cobb-Douglas.

The Intratemporal First-Order Condition

Within a period, the household chooses \(c\) and \(\ell\) to maximize utility subject to the spending constraint \(c + w \ell \leq \chi\). The FOC sets the marginal rate of substitution equal to the relative price:

\[\frac{u_\ell}{u_c} = w\]

For Cobb-Douglas preferences this reduces to:

\[\boxed{\; \frac{w \, \ell}{c} = \frac{\zeta}{1-\zeta} \;}\]

Leisure expenditure as a share of consumption is a constant determined by preferences alone.

Calibrating \(\zeta\)

Prescott pins down \(\zeta\) from long-run ratios. If average consumption equals average labor income (\(c \approx w \cdot n\)), then \(w \ell / c = \ell / n = \ell / (1 - \ell)\). Combining with the FOC:

\[\ell = \zeta\]

A 40-hour work week plus 8 hours of sleep a day leaves about two-thirds of waking time for leisure, so \(\zeta \approx 2/3\).

The preferred point in Prescott’s calibration: \(\zeta = 2/3\), \(\rho = 1\), \(\beta = 0.96\) (annual).

The Production Side

Output is Cobb-Douglas in capital and labor:

\[Y_t = A_t \, K_t^{1 - \alpha_L} \, N_t^{\alpha_L}\]

where \(\alpha_L\) is labor’s share in output, \(A_t\) is total factor productivity, and \(N_t\) is aggregate hours.

Capital accumulates with depreciation:

\[K_{t+1} = (1 - \delta) K_t + Y_t - C_t\]

Prescott calibrates \(\alpha_L = 0.64\) (labor share) and \(\delta = 0.10\) (annual depreciation).

RBC Dynamics: Labor and Leisure

The Productivity Process

Prescott uses the Solow residual as a measure of technology shocks. Defining \(\hat{X}_t \equiv (X_t - X_{t-1})/X_{t-1}\), the production function implies:

\[\hat{A}_t \approx \hat{Y}_t - \alpha_L \hat{N}_t - (1 - \alpha_L) \hat{K}_t\]

The residual \(\hat{A}_t\) is then modeled as:

\[\hat{A}_t = g_a + \varepsilon_t\]

with \(\sigma_\varepsilon = 0.0076\) per quarter. Shocks are large and persistent.

Combining the Two FOCs

The intertemporal Euler equation for consumption (separable utility, log case):

\[\frac{c_{t+1}}{c_t} = \beta \, R_{t+1}\]

The intratemporal FOC holds in both periods:

\[\frac{w_t \, \ell_t}{c_t} = \frac{\zeta}{1-\zeta} = \frac{w_{t+1} \, \ell_{t+1}}{c_{t+1}}\]

Rearranging and taking logs:

\[\boxed{\; \widehat{\ell}_{t+1} \approx -\widehat{w}_{t+1} + (r_{t+1} - \delta) \;}\]

How Hours Fluctuate

The leisure Euler equation says two forces can make hours move over the cycle:

Wage movements \(\widehat{w}_{t+1}\): if real wages rise in booms, workers substitute labor for leisure
Interest rate movements \(r_{t+1}\): if returns rise, workers shift effort forward to earn and invest

Both mechanisms are present in the model. The empirical question is whether either is quantitatively sufficient to reproduce the observed volatility of hours.

The Empirical Critique

Prescott’s Moment-Matching Table

Prescott (1986) summarizes the model’s performance with a three-column table:

Statistic	US Data	Model
\(\sigma_\ell\) (calibrated)	0.76	0.76
\(\sigma_y\) (output)	1.76	1.48
\(\sigma_n\) (hours)	1.67	0.76

The first column is calibrated, so it is not a test. The second column is respectable: the model generates output volatility that is 84 percent of the data. The third column is a disaster.

The Labor Volatility Puzzle

Hours in the model are less than half as volatile as hours in the data. Since the model was explicitly designed to match long-run trends in hours, the failure on business-cycle frequencies is sharp.

Something must make hours more responsive to the cycle than Prescott’s baseline allows. Two candidates emerge from the leisure Euler equation.

Mechanism One: Strong Labor Supply Response

Interpretation: transitory wage movements generate large hour responses.

Required: a large intertemporal elasticity of labor supply
Problem: micro studies estimate small elasticities, typically 0.1 to 0.5
Problem: wages do not obviously fall in recessions; the cyclicality of real wages is itself a debated empirical question

The evidence from household data does not support this mechanism.

Mechanism Two: Interest Rate Response

Interpretation: workers shift effort toward periods of high returns.

If this were the driver, the consumption Euler equation would imply:

\[\frac{c_{t+1}}{c_t} = \frac{w_{t+1} \, \ell_{t+1}}{w_t \, \ell_t}\]

With wages held fixed, consumption and leisure must move together. Recessions would be periods of high consumption and booms periods of low consumption.

The data show the opposite: consumption is strongly procyclical. The mechanism fails.

The Deeper Problem

Voluntary labor supply cannot generate realistic business cycle hours without either:

A counterfactually large labor supply elasticity, or
A counterfactual consumption pattern

This is a general issue for RBC models and motivated the development of New Keynesian DSGE, which adds nominal rigidities and allows for involuntary unemployment.

Visualizing the Puzzle

The leftmost pair is calibrated. The middle pair is a qualified success. The rightmost pair is the puzzle.

Synthesis

What DSGE Got Right, What It Got Wrong

Successes of the RBC program

Established a computable benchmark: everything in the model has a microfoundation
Showed that productivity shocks alone can generate sizable output fluctuations
Provided a laboratory for calibration exercises, moment matching, and policy experiments

Failures

Under-predicts hours volatility by a factor of two
Requires implausible labor supply mechanisms
Ignores frictions (sticky prices, search, credit constraints) that later literatures show matter

Bridge to Modern DSGE

New Keynesian DSGE models address the RBC critique by adding:

Friction	Consequence
Sticky prices (Calvo)	Monetary policy has real effects
Sticky wages (Taylor)	Unemployment responds to demand
Search frictions	Involuntary unemployment is possible
Borrowing constraints	Consumption becomes more procyclical

These extensions preserve the Euler equation at the center of the model while relaxing the frictionless price-taking assumption. Modern central bank models (Smets-Wouters, FRB/US) are direct descendants.

Key Takeaways (1/2)

Brock-Mirman delivers a closed-form MPC of \(\kappa = 1 - \alpha \beta\) only because of three knife-edge assumptions: log utility, 100% depreciation, and Cobb-Douglas production
The Euler equation extends naturally to general equilibrium; the analytical difficulty comes from the budget constraint and the state space, not the optimality conditions
Prescott RBC adds labor-leisure choice, requiring Cobb-Douglas utility to match the long-run trend in hours and calibrating \(\zeta\) from the time budget

Key Takeaways (2/2)

Prescott’s productivity process is a random walk in the Solow residual; the model matches output volatility but under-predicts hours volatility by half
The labor volatility puzzle can only be resolved by either counterfactually large labor supply elasticities or counterfactual consumption comovement
This failure motivated New Keynesian DSGE: same Euler equation, but frictions on prices, wages, and credit generate realistic hours dynamics