We are going to study a textbook neoclassical growth model in the lab (see Li and Noussair, AER, 2000).
where $\rho$ is the discount rate, $u(.)$ is a concave utility function, $k_t$ is the stock of capital, $f$ is the production function, and $\delta$ is depreciation.
Let's find the value function,
\begin{equation} V(k_{t}) = \max_{c_t, k_{t+1}}\{u(c_t)+(1+\rho)^{-t} E[V(k_{t+1})] \} \end{equation}\begin{equation} V(k_{t}) = \max_{k_{t+1}}\{u(f(k_t) + (1-\delta) k_{t}-k_{t+1})+(1+\rho)^{-1} E[V(k_{t+1})] \} \end{equation}FOC \begin{equation} u'(c_t) = (1+\rho)^{-1} E_t[V_k(k_{t+1})] \end{equation}
Envelop theorem \begin{equation} V_k(k_t) = u'(c_t)(1-\delta+f') \end{equation}
Envelop condition \begin{equation} E_t [V_k(k_{t+1})] = E_t[u'(c_{t+1})(1-\delta+f') ] \end{equation}
Stochastic Euler equation
\begin{equation} u'(c_t) = (1+\rho)^{-1} E_t[u'(c_{t+1})(1-\delta+f') ] \end{equation}Steady-state $c_{t+1}=c_t=\bar{c}$ and $k_{t+1}=k_t=\bar{k}$ \begin{equation} f' = \rho + \delta \end{equation} \begin{align} \bar{c}=f(\bar{k})-\delta \bar{k} \end{align}
from IPython.core.display import HTML
def css_styling():
styles = open("custom.css", "r").read()
return HTML(styles)
css_styling()