\documentclass[a4paper,11pt]{article}
\title{Monetary Business Cycle Model}
\author{Eric Scheffel}
\begin{document}

\maketitle


The problem is set up as a social planner's problem in the following way:

\begin{eqnarray*}
L & = & E_0\sum_{t=0}^\infty\beta^t\Biggl\lbrace\frac{\left[c_t\left(1-l_t\right)^A\right]^{\left(1-\eta\right)}}{1-\eta}\\
  & + & \lambda_t\Bigl[w_tl_t+r_t^Kk_{t-1}+\left(1-\delta\right)k_{t-1}+\frac{1}{1-\pi_t}m_{t-1}+\frac{1+i_t}{1-\pi_t}b_{t-1}\\
  &   & -m_t-b_t-c_t-k_t\Bigr]\\
  & + & \Bigl[\frac{1}{1+\pi_t}m_{t-1}+\tau_t-c_t\Bigr]\Biggr\rbrace
\end{eqnarray*}
The first-order conditions to this problem are as follows:
\begin{eqnarray*}
c_t: & & c_t^{-\eta}\left(1-l_t\right)^{A\left(1-\eta\right)}-\lambda_t-\mu_t = 0;\\
l_t: & & Ac_t^{1-\eta}\left(1-l_t\right)^{A\left(1-\eta\right)-1}-\lambda_tw_t = 0;\\
k_t: & & \beta\lambda_{t+1}\left(r_{t+1}^K+1-\delta\right)-\lambda_t = 0;\\
m_t: & & \frac{\beta\lambda_{t+1}}{1+\pi_{t+1}}+\frac{\beta\mu_{t+1}}{1+\pi_{t+1}}-\lambda_t = 0;\\
b_t: & & \beta\frac{1+i_{t+1}}{1+\pi_{t+1}}-\lambda_t = 0;
\end{eqnarray*}
Notice that the FOCs for capital,money and bonds, imply the following:
\[\frac{\lambda\left(1+\pi_{t+1}\right)}{\beta\lambda_{t+1}}=1+\frac{\mu_{t+1}}{\lambda_{t+1}}=1+i_{t+1}=\left(1+r_{t+1}^K-\delta\right)\left(1+\pi_{t+1}\right)\]
or
\[1+\frac{\mu_{t+1}}{\lambda_{t+1}}=1+i_{t+1}=\left(1+r_{t+1}\right)\left(1+\pi_{t+1}\right)\]
which is a Fisher equation type relationship. Also, this means that the money and bond FOCs are really only helpful
in deriving this Fisherian relationship. We can re-write the FOCs as follows:
\[c_t^{-\eta}\left(1-l_t\right)^{A\left(1-\eta\right)}-\lambda_t\left(1+r_t\right)\left(1+\pi_t\right) = 0;\]
\[Ac_t^{1-\eta}\left(1-l_t\right)^{A\left(1-\eta\right)-1}-\lambda_tw_t = 0;\]
\[\beta\lambda_{t+1}\left(r_{t+1}^K+1-\delta\right)-\lambda_t = 0;\]
or, using gross rates and eliminating consumption, using money:
\[m_t^{-\eta}\left(1-l_t\right)^{A\left(1-\eta\right)}-\lambda_tR_t\Pi_t = 0;\]
\[Am_t^{1-\eta}\left(1-l_t\right)^{A\left(1-\eta\right)-1}-\lambda_tw_t = 0;\]
\[\beta\lambda_{t+1}R_{t+1}-\lambda_t = 0;\]
We can therefore write down the following non-linear system of equations:
\begin{equation}
z_tk_t^{\rho}l_t^{1-\rho}-m_{t+1}-k_{t+1}+\left(1-\delta\right)k_t = 0;
\end{equation}
\begin{equation}
\frac{1-l_t}{Ac_t}-\frac{R_t\Pi_t}{w_t} = 0;
\end{equation}
\begin{equation}
\frac{U_{1,t}}{R_t\Pi_t}-\frac{E_tU_{1,t+1}}{E_t\Pi_{t+1}} = 0;
\end{equation}
where we have that:
\[U_{1,t} = m_{t+1}^{-\eta}\left(1-l_t\right)^{A\left(1-\eta\right)}\]
\[E_tU_{1,t+1} = \left[E_tm_{t+2}\right]^{-\eta}\left(1-E_tl_{t+1}\right)^{A\left(1-\eta\right)};\]
\[E_tm_{t+2} = \left(1+\left(\Theta+E_tu_{t+1}-1\right)\right)m_{t+1}/E_t\Pi_{t+1} \]
continuing with the system of equations \ldots
\begin{equation}
m_{t+1}-\left(1+\left(\Theta+u_t-1\right)\right)\frac{m_t}{\Pi_t} = 0;
\end{equation}
\begin{equation}
m_{t+1}-c_t = 0;
\end{equation}
\begin{equation}
E_tz_{t+1}-z^{1-\psi_z}z_t^{\psi_z}\epsilon_{z,t} = 0;
\end{equation}
\begin{equation}
E_tu_{t+1}-u^{1-\psi_u}u_t^{\psi_u}\epsilon_{u,t} = 0;
\end{equation}

\end{document}