notes.tex

\documentclass[10pt]{article}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}

% partial derivative as a fraction
\newcommand{\fracpd}[2]{
  \ensuremath{\frac{\partial #1}{\partial #2}}
}


\begin{document}

% --------------------------------------- %
% Section
% --------------------------------------- %
\section{Introduction}

  This repository holds the solutions to a general set of growth models that incorporate the following explore the differences between

  \begin{itemize}
    \item One Agent vs Two Agent
    \item One Good vs Two Goods
    \item Time Additive Preferences vs Recursive Preferences
    \item Constant Volatility vs Stochastic Volatility
  \end{itemize}

  By solve the 16 possible versions of this model, we are able to isolate the effects of each bell and whistle that we include in the model.

% --------------------------------------- %
% Section
% --------------------------------------- %
\section{One Agent One Good General Model}

  We will write our one agent general unscaled model as the following:

  \begin{align*}
    J_t(k_t, x_t, v_t) &= \max_{k_{t+1}} \left[(1 - \beta) c_t^{\rho} + \beta \mu(J_{t+1}(k_{t+1}, x_{t+1}, v_{t+1}))^{\rho} \right]^{\frac1\rho} \\
    \text{Subject To }& \\
    c_t &= \left[ \eta k_{t}^{\nu} + (1 - \eta) z_t^{\nu} \right]^{\frac1\nu} + (1 - \delta) k_t - k_{t+1} \\
    x_{t+1} &= A x_{t} + B v_t^{\frac{1}{2}} \varepsilon_{1, t+1} \\
    v_{t+1} &= (1 - \phi_v) \bar{v} + \phi_v v_{t} + \tau \varepsilon_{2, t+1} \\
    \log(z_t) &= \log(z_{t-1}) + \log(\bar{g}) + x_t
  \end{align*}

  We could then scale the model by dividing by $z_t$ to give us (will not both writing tildes for now -- everything here is divided by $z_t$)

  \begin{align*}
    J_t(k_t, x_t, v_t) &= \max_{\chi_{t}} \left[(1 - \beta) c_t^{\rho} + \beta \mu(J_{t+1}(k_{t+1}, x_{t+1}, v_{t+1}) g_{t+1})^{\rho} \right]^{\frac1\rho} \\
    \text{Subject To }& \\
    c_t &= \left[ \eta k_{t}^{\nu} + (1 - \eta) \right]^{\frac1\nu} + (1 - \delta) k_t - \chi_t \\
    \chi_t &= k_{t+1} g_{t+1} \\
    x_{t+1} &= A x_{t} + B v_{t}^{\frac{1}{2}} \varepsilon_{1, t+1} \\
    v_{t+1} &= (1 - \phi_v) \bar{v} + \phi_v v_{t} + \tau \varepsilon_{2, t+1} \\
    \log(g_t) &= \log(z_t) - \log(z_{t-1}) = \log(\bar{g}) + x_t
  \end{align*}

  Notice we can get constant volatility by allowing $\tau = \phi_v = 0$ or time separable preferences by $\rho = \alpha$.

% --------------------------------------- %
% Section
% --------------------------------------- %
\section{One Agent Two Goods General Model}

% --------------------------------------- %
% Section
% --------------------------------------- %
\section{Two Agent One Good General Model}

% --------------------------------------- %
% Section
% --------------------------------------- %
\section{Two Agent Two Goods General Model}


\end{document}