Chapter 17: Recursive Methods for Real Business Cycle Models

Value Function Iteration over the (k, A) State Space

“The RBC model is a quantitative theory machine: feed in calibrated parameters, compute the equilibrium policy functions, simulate the model, and compare the resulting second moments to the data.” — Finn Kydland and Edward Prescott

Cross-reference: Principles Ch. 27 (RBC model: specification, calibration, stylized facts, criticisms) [P:Ch.27]

17.1 The RBC Model as a Recursive Problem¶

The Real Business Cycle (RBC) model of Kydland and Prescott (1982) and Long and Plosser (1983) is the discrete-time stochastic extension of the neoclassical growth model. It adds:

1. Stochastic TFP: Technology $A_t$ follows an AR(1) process in logs.

2. Labor-leisure choice: Households choose hours worked as well as consumption and saving.

3. Competitive equilibrium: Prices are determined in competitive markets; the equilibrium can be decentralized.

The key insight of Kydland and Prescott: the competitive equilibrium of the RBC model is equivalent to the solution of a social planner’s problem (by the First Welfare Theorem, since there are no externalities or market failures). This means we can solve the planner’s problem using the dynamic programming tools of Chapter 15 and interpret the solution as the competitive equilibrium.

The state variables of the RBC model are $(k_t, A_t)$: the current capital stock and the current TFP level. Both are known at the beginning of period $t$; both are inherited from the past. The control variables are $(c_t, n_t)$: consumption and hours worked chosen at the beginning of period $t$.

17.2 Model Specification¶

17.2.1 Preferences¶

The representative household maximizes:

where $n_t \in [0,1]$ is hours worked, $1-n_t$ is leisure, $\mu \in (0,1)$ is the consumption share in the composite good, and $\sigma > 0$ is the curvature parameter (the coefficient of relative risk aversion over the composite good).

This King-Plosser-Rebelo (1988) utility specification is consistent with balanced growth: with log-utility in the composite good ($\sigma = 1$), the hours-worked ratio $n^*$ is constant along the balanced growth path — consistent with the Kaldor fact of stable labor supply [P:Ch.5.1].

17.2.2 Production and Capital Accumulation¶

17.2.3 TFP Process¶

with $\rho_A \in (0,1)$ and $\sigma_A > 0$. The TFP shock $\varepsilon_t^A$ is the fundamental driver of business cycles in the RBC model.

17.2.4 The Planner’s Problem¶

The social planner maximizes household utility subject to feasibility and the TFP process. The state at the beginning of period $t$ is $\mathbf{s}_t = (k_t, A_t)$. The planner’s Bellman equation:

subject to:

The expectation $\mathbb{E}_A[V(k', A')]$ is over the next period’s TFP realization, conditional on the current $A$.

17.3 Value Function Iteration over the $(k, A)$ Grid¶

17.3.1 State Space Discretization¶

Capital grid: $\mathcal{K} = \{k_1, \ldots, k_{N_k}\}$ with $k_1 = k_{min} \approx (1-\varepsilon)k^*$ and $k_{N_k} = k_{max} \approx (1+\varepsilon)k^*$ for some $\varepsilon > 0$ (e.g., $\varepsilon = 0.5$ means ±50% around the steady state).

TFP grid: $\mathcal{A} = \{A_1, \ldots, A_{N_A}\}$ discretized from the AR(1) using Tauchen (1986) with $N_A$ points and transition matrix $\Pi \in \mathbb{R}^{N_A\times N_A}$.

The value function is represented as an $N_k \times N_A$ matrix $\mathbf{V}$, with $V_{ij} \approx V(k_i, A_j)$.

17.3.2 The Bellman Operator in Matrix Form¶

For each $(k_i, A_j)$:

1. Compute output: $Y_{ij}^n = A_j k_i^\alpha n^{1-\alpha}$ for each feasible $n$.

2. Compute consumption: $c = Y_{ij}^n + (1-\delta)k_i - k'$ for each feasible $(n, k')$.

3. Compute utility: $u(c, 1-n)$ if $c > 0$, else $-\infty$.

4. Compute expected next-period value: $\mathbb{E}_j[V(k')] = \sum_{j'}\Pi_{jj'}V_{k',j'}$.

5. Maximize over $(n, k')$ to get $V^{new}_{ij}$.

Theorem 17.1 (VFI Convergence for the RBC Model). Under standard regularity conditions on $u$ (continuous, strictly concave, satisfying Inada conditions), the Bellman operator for the RBC model is a contraction with modulus $\beta$. VFI converges to the unique fixed point.

Proof. Same structure as Theorem 15.2, extended to the two-dimensional state space. The key step: Blackwell’s monotonicity and discounting conditions are satisfied because $u$ is increasing in $c$, $V$ is evaluated at the same $\beta$ in each iteration, and the constraint set is compact. $\square$

17.3.3 Simplification: Analytical Labor Optimality¶

For the KPR utility specification with $\sigma = 1$ (log in composite):

the labor-leisure optimality condition is:

This pins down $n_t$ as a function of $(k_t, A_t, c_t)$. Substituting into the resource constraint, we can eliminate $n$ and solve the remaining $(k, c)$ problem — reducing the dimension of the optimization by one. This computational trick speeds up VFI substantially.

For general $\sigma$, we solve for $n$ numerically at each grid point using the first-order condition above.

17.4 Calibration: The Kydland–Prescott Method¶

Definition 17.1 (Calibration). Calibration is the process of assigning numerical values to model parameters by: (i) matching parameters that have direct empirical counterparts (e.g., $\alpha$ = capital share in national income, $\delta$ = depreciation rate); (ii) setting preference parameters to match long-run ratios (e.g., $\mu$ set so that $n^* = 0.33$, one-third of time at work).

Standard RBC calibration (quarterly data):

Steady-state relations used in calibration:

From the Euler equation at the steady state: $1 = \beta(1 + r^* - \delta) \Rightarrow r^* = 1/\beta - 1 + \delta$.

From the capital-output ratio: $K^*/Y^* = \alpha/(r^*+\delta)$.

From labor market clearing: $(1-\mu)/\mu = (1-\alpha)Y^*/c^* \cdot (1-n^*)/n^*$, pins down $\mu$ given $n^* = 1/3$.

17.5 The HP Filter and Business Cycle Moments¶

Macroeconomic data contain trends (long-run growth) and cycles (short-run fluctuations). The Hodrick–Prescott filter [P:Ch.6.3] separates them by solving:

where $y_t$ is log output, $\tau_t$ is the trend component, and $\lambda = 1600$ (for quarterly data). The cycle is $c_t^{HP} = y_t - \tau_t$.

In matrix form, the solution is $\boldsymbol\tau = (I + \lambda D'D)^{-1}\mathbf{y}$, where $D$ is the second-difference matrix. In APL, this is:

Target second moments (U.S. quarterly data, HP-filtered, 1954–2019):

The RBC model must match these moments — particularly the relative volatility of consumption (less volatile than output) and investment (more volatile), and the high contemporaneous correlation of all variables with output.

17.6 Full VFI Implementation¶

R¶

APL¶

Python¶

Julia¶

# Julia — RBC model: Tauchen + VFI essentials
using Statistics, Distributions, Interpolations

alpha, delta, beta, mu = 0.36, 0.025, 0.99, 0.357
rho_A, sig_A = 0.95, 0.0072
r_star = 1/beta - 1 + delta
kstar  = (alpha/(r_star+delta))^(1/(1-alpha))
n_opt  = 1/3  # simplified constant labor

Nk, NA = 80, 7
k_grid = range(0.5*kstar, 1.5*kstar, length=Nk) |> collect
sig_z  = sig_A/sqrt(1-rho_A^2)
z_grid = range(-3sig_z, 3sig_z, length=NA) |> collect
A_grid = exp.(z_grid)

# Tauchen transition matrix
d = Normal(0, sig_A); dz = step(range(-3sig_z,3sig_z,length=NA))
Pi = zeros(NA, NA)
for i in 1:NA
    mu_z = rho_A*z_grid[i]
    Pi[i,1]    = cdf(d, z_grid[1]+dz/2-mu_z)
    Pi[i,NA]   = 1 - cdf(d, z_grid[NA]-dz/2-mu_z)
    for j in 2:NA-1
        Pi[i,j] = cdf(d, z_grid[j]+dz/2-mu_z) - cdf(d, z_grid[j]-dz/2-mu_z)
    end
end

util(c, n) = mu*log(max(c,1e-10)) + (1-mu)*log(max(1-n,1e-10))

V = zeros(Nk, NA)
println("Running RBC VFI...")
for iter in 1:200
    EV   = V * Pi'   # Nk × NA: expected continuation
    V_new = fill(-Inf, Nk, NA)
    for j in 1:NA, i in 1:Nk
        Aj, ki = A_grid[j], k_grid[i]
        x = Aj*ki^alpha*n_opt^(1-alpha) + (1-delta)*ki
        best = -Inf
        for ip in 1:Nk
            kp = k_grid[ip]; c = x - kp
            c > 0 || continue
            pay = util(c, n_opt) + beta*EV[ip,j]
            pay > best && (best = pay)
        end
        V_new[i,j] = best
    end
    diff = maximum(abs.(V_new - V)); V .= V_new
    iter % 25 == 0 && println("  iter=$iter, diff=$(round(diff,digits=6))")
    diff < 1e-5 && (println("Converged at iter $iter"); break)
end
println("VFI complete. V range: [$(round(minimum(V),digits=2)), $(round(maximum(V),digits=2))]")

17.7 Programming Exercises¶

Exercise 17.1 (APL — HP Filter)¶

Implement the HP filter in APL using ⌹ (matrix divide) to solve the linear system $(I + \lambda D'D)\bm\tau = \mathbf{y}$. (a) Construct the second-difference matrix $D$ using outer products and diagonal indexing. (b) Apply to simulated AR(1) GDP data with $\lambda = 1600$. (c) Verify that the cycle component has the correct spectral properties (band-pass behavior around business cycle frequencies 6–32 quarters).

Exercise 17.2 (Python — Second Moments)¶

Using the VFI solution from Section 17.6: (a) simulate 10,000 periods of the RBC model; (b) compute HP-filtered second moments for $(Y, C, I, n)$; (c) compare to the data targets in Section 17.5. What are the model’s main failures? (The standard RBC model generates too-correlated labor with output and too-high investment volatility relative to data.)

Exercise 17.3 (Julia — Calibration Sensitivity)¶

For the RBC calibration, vary $\rho_A \in \{0.7, 0.8, 0.9, 0.95, 0.99\}$ and $\sigma_A \in \{0.005, 0.007, 0.010, 0.015\}$. For each combination: (a) simulate the model for 1,000 periods; (b) compute the output standard deviation and the consumption-to-output relative volatility; (c) identify which combination best matches U.S. data moments. Plot the results as a heat map.

Exercise 17.4 — Log-Linearization Preview ($\star$)¶

Log-linearize the RBC equilibrium conditions around the non-stochastic steady state (following Chapter 27’s approach): (a) the Euler equation; (b) the labor optimality condition; (c) the resource constraint; (d) the capital accumulation equation. Write the log-linearized system in the state-space form $\hat{\mathbf{y}}_{t+1} = A\hat{\mathbf{y}}_t + B\hat\varepsilon_{t+1}$. (e) Compute the eigenvalues of $A$ and verify the Blanchard–Kahn condition (one stable eigenvalue for one predetermined variable $\hat{k}_t$).

Exercise 17.5 — Endogenous Labor ($\star\star$)¶

Extend the VFI implementation to include endogenous labor supply: for each $(k_i, A_j, k'_m)$ triple, solve the household’s labor optimality condition $[(1-\mu)/mu] \cdot c / (1-n) = (1-\alpha)Y/n$ jointly with the resource constraint $c + k' = Y + (1-\delta)k$. (a) Implement as an inner optimization loop using scipy’s brentq. (b) Compute the policy functions $c^*(k,A)$ and $n^*(k,A)$. (c) Show that endogenous labor amplifies the output response to TFP shocks (the labor supply channel).

17.8 Chapter Summary¶

Key results:

• The RBC model is the stochastic extension of the Ramsey growth model with endogenous labor, solved as a social planner’s problem using VFI over the $(k, A)$ state space.

• The state space is $(k_t, A_t)$; the TFP process is discretized to a Markov chain via Tauchen’s method, yielding the $N_A\times N_A$ transition matrix $\Pi$.

• The Bellman equation $V(k,A) = \max_{c,n,k'}\{u(c,1-n)+\beta\mathbb{E}[V(k',A')]\}$ is solved by VFI; convergence is guaranteed by Theorem 17.1 (same as Theorem 15.2, extended to 2D).

• Calibration uses national accounts (capital share $\alpha$, depreciation $\delta$), asset-return data (discount factor $\beta$), and the Solow residual (TFP parameters $\rho_A$, $\sigma_A$). The preference parameter $\mu$ is set to match a labor supply target.

• The HP filter $(I+\lambda D'D)^{-1}$ separates trend from cycle; in APL it is implemented as ⌹(I+lambda×(⍉D)+.×D)+.×y.

• Second moments (standard deviations and correlations of HP-filtered variables) are the primary targets for model evaluation. The standard RBC model matches output variance and consumption relative volatility reasonably well, but overpredicts labor volatility.

Connections forward: Chapter 27 (log-linearization) provides an analytic approximation to the VFI policy functions, dramatically reducing computation time. Chapter 28 (Blanchard–Kahn) applies the determinacy framework to verify the RBC model has a unique stable solution. Chapter 37 (Great Recession replication) adds financial frictions to the RBC structure and estimates the augmented model on data.

Next: Part V — Stochastic Methods for Macroeconomic Modeling