An RBC model with two correlated shocks

Computational Macro Cookbook · Beyond Research

Households choose consumption and hours, firms produce with capital and labor, and capital is built from investment. Two shocks hit this economy: a technology shock \(a_t\) that makes production more productive, and an investment-specific shock \(b_t\) that makes each unit of investment add more capital. The two shocks are correlated.

\[\begin{aligned} \text{Preferences:}\quad & E_0 \textstyle\sum_{t \ge 0} \beta^t \Big( \log c_t - \theta \, \dfrac{h_t^{1+\psi}}{1+\psi} \Big) \\ \text{Production:}\quad & y_t = e^{a_t} \, k_t^{\alpha} \, h_t^{1-\alpha} \\ \text{Resources:}\quad & c_t + i_t = y_t \\ \text{Capital:}\quad & k_{t+1} = e^{b_t} \, i_t + (1-\delta) \, k_t \\ \text{Shocks:}\quad & a_t = \rho \, a_{t-1} + \tau \, b_{t-1} + \varepsilon_t \\ & b_t = \tau \, a_{t-1} + \rho \, b_{t-1} + \nu_t, \qquad \operatorname{corr}(\varepsilon_t, \nu_t) = \varphi \end{aligned}\]

The technology innovation \(\varepsilon_t\) enters production through \(a_t\); the investment-specific innovation \(\nu_t\) enters capital accumulation through \(b_t\). These are the shocks you control below.

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References

Collard, F. (2001). Stochastic simulations with Dynare: A practical guide. Adapted for Dynare 4.1 by M. Juillard and S. Villemot (2009). CEPREMAP.
Collard, F. and M. Juillard (2001). Accuracy of stochastic perturbation methods: The case of asset pricing models. Journal of Economic Dynamics and Control 25, 979–999.
Schmitt-Grohé, S. and M. Uribe (2004). Solving dynamic general equilibrium models using a second-order approximation to the policy function. Journal of Economic Dynamics and Control 28, 755–775.
Dynare, version 6.3. www.dynare.org/resources