# RBC Model

#### Background

This tool simulates the basic RBC model.: \begin{align} \varphi\left(1-L_t\right)^{-\eta} & = C_t^{-\sigma} W_t \tag{1}\\ C_t^{-\sigma} & = \beta E_t \left[C_{t+1}^{-\sigma}R_{t+1} + 1 - \delta)\right] \tag{2}\\ W_t & = (1-\alpha)A_tK_t^{\alpha}L_t^{-\alpha} \tag{3}\\ R_t & = \alpha A_tK_t^{\alpha-1}L_t^{1-\alpha} \tag{4}\\ K_{t+1} & = I_t + (1-\delta)K_t \tag{5}\\ Y_t & = C_t + I_t\tag{6}\\ Y_t & = A_t K_t^{\alpha}L_t^{1-\alpha} \tag{7}\\ \log A_{t+1} & = \rho_a \log A_{t} + \epsilon_{t+1} \tag{8} \end{align} where $$\epsilon_{t+1}\sim\mathcal{N}(0,\sigma_{\epsilon}^2)$$. Equation (1) is the representative household's first-order condition for the optional choice of labor. Equation (2) is the household's Euler equation reflecting an optimal choice of capital for period $$t+1$$. Equations (3) and (4) are the firm sector's first-order conditions for optimal choices of labor and capital. Equations (5), (6), and (7) describe the evolution of the aggregate capital stock, the goods market clearing condition, and the production function. Finally, equation (8) indicates that log TFP follows an AR(1) process.

The tool constructs a log-linear approximation of the equilibrium conditions and solves for the equilibrium values of the endogenous variables in terms of the state variables $$A_t$$ and $$K_t$$. To learn more, please see my notes on the solution method.

#### Instructions

Choose the values the parameters. Then choose the number of periods for the simulation and select which type of simulation that you wish to perform:

1. Stochastic simulation: a new random value for the white noise process $$\epsilon_t$$ will be drawn for each period.
2. Impulse response: $$\epsilon_t$$ will be set to $$\sigma_{\epsilon}$$ in period 1 and to 0 in all subsequent periods.
Simulated data are in units of log-deviations of each variable from its steady state. Use the dropdown menu in the upper-right corner of each plot to download the simulated data in csv format.

Simulation Settings

Parameters