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\begin{document}
\begin{flushright}
Jaime Frade\\
ECO5282-Dr. Garriga\\
Homework\:$\#2$\\
\end{flushright}
\begin{enumerate}
\item A economy consists of two infintely lived consumers named $i=1,2$.  There is one nonstorable consumption good.  Consumer $i$ consumers $c^i$ at time $t$.  Consumer $i$ ranks consumptions streams by $\sum^\infty_{t=0}\,\beta^t u(c^i_t)$ where $\beta\in(0,1)$ and $u(c)$ is strictly increasing, concave, and twice continiously differentiable.  Comsumer $1$ is endowed with a stream of the consumption good $y^1_t = 1,0,0,1,0,0,1, \ldots$.  Consumer 2 is endowed with a stream of the consumption good $y^1_t = 0,1,1,0,1,1,0, \ldots$.  Assume that markets are complete with time-0 trading.
\begin{description}
\item[(a)]  Define a competitive equilibrium.  
\vspace{.25cm}
\\
\textbf{Definition} An \underline{allocation} for agent $i$ is defined as state contingent function $c^i = \{c^i_t(s^t)\}^\infty_{t=0}$ for $i=1,2$
\vspace{.25cm}
\\
\textbf{Definition} A allocation is said to be a \underline{feasible allocation} if it satisfies
\begin{equation}\label{eq:feas1}
\sum^2_{i=1} c^i_t(s^t) = \sum^2_{i=1} y^i_t(s_t)
\end{equation}

\textbf{Definition} A \underline{competitive equilibrium} is a feasible allocation, $\{c^i\}^2_{i=1} = \{\{c^i_t(s^t)\}^\infty_{t=0}\}^2_{i=1}$, and a price system, $\{p^0_t(s^t)\}^\infty_{t=0}$, such that the allocation solves each household problem.
\vspace{.25cm}
\\
For a given household $i$ solves 
\begin{displaymath}
U(c^i) = \max_{\{c^T_t(s^t)\}} E_0 \sum^\infty_{t=0}\beta^t\,u(c^i_t)
\end{displaymath}
\begin{equation}\label{eq:max1}
U(c^i) = \max_{\{c^T_t(s^t)\}}\sum^\infty_{t=0}\sum_{s^t}\beta^t\,\pi(s^t|s_0)\,u(c^i_t(s^t)) 
\end{equation}

\begin{equation}\label{eq:bc1}
\textrm{subject to} \hspace{.25cm} \sum^\infty_{t=0}\sum_{s^t}\,p^0_t(s^t)c^i_t(s^t) = \sum^\infty_{t=0}\sum_{s^t}\,p^0_t(s^t)y^i_t(s_t), \hspace{.25cm} \textrm{and} \hspace{.25cm} c^1_t(s^t),c^2_t(s^t) \geq 0 
\end{equation}

\textbf{First Order Conditions:}
\vspace{.25cm}
\\
$\beta^t\,\pi(s^t|s_0)\,u\,'(c^i_t(s^t)) = \gamma^i\,p^0_t(s^t)$
\vspace{.25cm}
\\
\begin{equation} \label{eq:max2}
p^0_t(s^t) = \beta^t\,\pi(s^t|s_0)\,\dfrac{u\,'(c^i_t(s^t))}{\gamma^i}
\end{equation}

at $t=0$, we have the following:
\vspace{.25cm}
\\
$p\,^0_0(s^0) = \dfrac{u\,'(c^i_0(s^0))}{\gamma^i}$
\vspace{.25cm}
\\
solving for $p^0_0(s^0) = 1$, we have the following:
\vspace{.25cm}
\\
$1 = \dfrac{u\,'(c^i_0(s^0))}{\gamma^i}$ \hspace{.25cm} $\Rightarrow$ \hspace{.25cm} $\gamma^i = u\,'(c^i_0(s^0))$
\vspace{.25cm}
\\
Subsititing into (\ref{eq:max2}), obtain
\begin{equation} \label{eq:max3}
p^0_t(s^t) = \beta^t\,\pi(s^t|s_0)\,\dfrac{u\,'(c^i_t(s^t))}{u\,'(c^i_0(s^0))}
\end{equation}

For the problem above for $i=1,2$, there is no aggregate risk (hence $c^i_t = c^i_0$), thus
\vspace{.25cm}
\\
$\beta^t\,\pi(s^t|s_0)\,\dfrac{u\,'(c^1_t(s^t))}{u\,'(c^1_0(s^0))} = p^0_t(s^t) = \beta^t\,\pi(s^t|s_0)\,\dfrac{u\,'(c^2_t(s^t))}{u\,'(c^2_0(s^0))}$
\vspace{.25cm}
\\

\newpage

In equilibrium, from $MRS^1_{0,T} = MRS^2_{0,T}$ obtains
\vspace{.25cm}
\\
\begin{equation}\label{eq:equal}
\dfrac{u\,'(c^1_t(s^t))}{u\,'(c^1_0(s^0))} =\dfrac{u\,'(c^2_t(s^t))}{u\,'(c^2_0(s^0))}
\end{equation}

Subsituting into the buget constraints (\ref{eq:bc1}) to find the feasible allocation for each agent for a competitive equilibrium.
\vspace{.25cm}
\begin{displaymath}
\sum^\infty_{t=0}\sum_{s^t}\,\beta^t\,\pi(s^t|s_0)\,\dfrac{u\,'(c^i_0(s^t))}{\gamma^i}\,[c^i_t(s^t) - y^i_t(s_t)] = 0
\end{displaymath}
\begin{displaymath}
\dfrac{u\,'(c^i_0(s^t))}{\gamma^i}\sum^\infty_{t=0}\sum_{s^t}\,\beta^t\,\pi(s^t|s_0)\,[c^i_t(s^t) - y^i_t(s_t)] = 0
\end{displaymath}

Because $\dfrac{u\,'(c^i_0(s^t))}{\gamma^i} \neq 0$, then obtain the following to solve for $c^i_0$
\begin{displaymath}
c^i_o\,\underbrace{\sum^\infty_{t=0}\beta^t}_{=\frac{1}{1-\beta}} \,\, \underbrace{\sum_{s^t}\pi(s^t|s_0)}_{=1} = \sum^\infty_{t=0}\sum_{s^t}\,\beta^t\,\pi(s^t|s_0)\,y^i_t(s_t)
\end{displaymath}
Sum of probabilities in all states, $\pi(s^t|s_0) = 1$, and using a geometric series for for $\beta$
\begin{displaymath}
\dfrac{u\,'(c^i_0(s^t))}{\gamma^i}\sum^\infty_{t=0}\sum_{s^t}\,\beta^t\,\pi(s^t|s_0)\,[c^i_t(s^t) - y^i_t(s_t)] = 0
\end{displaymath}
\begin{equation}\label{eq:consum}
c^i_0 = (1-\beta)\,\sum^\infty_{t=0}\sum_{s^t}\,\beta^t\,\pi(s^t|s_0)\,y^i_t(s_t)
\end{equation}

Solve each consumption for each agent, $i=1,2$, where comsumer $1$ is endowed with a stream of the consumption good $y^1_t = 1,0,0,1,0,0,1, \ldots$.  Consumer 2 is endowed with a stream of the consumption good $y^1_t = 0,1,1,0,1,1,0, \ldots$.
\vspace{.25cm}
\\
\begin{tabular}{l l l}
\textbf{agent 1}	& \hspace{.25cm} & \hspace{.25cm} \textbf{agent 2}
\vspace{.5cm}
\\
$c^1 = (1-\beta) \sum^\infty_{t=0}\, \beta^t\, y^1_t$ & \hspace{.25cm} & \hspace{.25cm} $c^2 = (1-\beta) \sum^\infty_{t=0}\, \beta^t\, y^2_t$
\vspace{.5cm}
\\
$c^1 = (1-\beta) [\beta^0 + \beta^3 + \beta^6 + \ldots]$ & \hspace{.25cm} & \hspace{.25cm} $c^2 = (1-\beta) [\beta + \beta^2 + \beta^4 + \beta^5 + \ldots]$
\vspace{.5cm}
\\
$c^1 = (1-\beta) \sum^\infty_{t=0} \beta^{3t}$ & \hspace{.25cm} & \hspace{.25cm} $c^2 = (1-\beta) \sum^\infty_{t=0} [\beta^{t} - \beta^{3t}]$
\vspace{.5cm}
\\
using sum of a geometric series, can obtain:
\vspace{.25cm}
\\
$c^1 = \dfrac{(1-\beta)}{(1-\beta^3)} = \dfrac{1}{1+\beta+\beta^2}$ & \hspace{.25cm} & \hspace{.25cm} $c^2 = \dfrac{(1-\beta)}{(1-\beta)} - \dfrac{(1-\beta)}{(1-\beta^3)}= \dfrac{\beta\,(1+\beta^2)}{1+\beta+\beta^2}$
\end{tabular}

\begin{eqnarray}\label{eq:comsum}
c^1 = \dfrac{1}{1+\beta+\beta^2} \hspace{.25in} \textrm{and} \hspace{.25in} c^2 = \dfrac{\beta\,(1+\beta^2)}{1+\beta+\beta^2}
\vspace{.33in}
\\
\textrm{subject to:} \hspace{.25cm} \sum^2_1 c^i = w_0  = 1 \Rightarrow c^1 + c^2 = 1 \label{eq:sbto1}
\end{eqnarray}



\newpage

\item[(b)]  Compute a competitive equilibrium.  
\vspace{.25cm}
\\
Since only one agent, \textbf{comsumption $=$ endowment}, obtain the following:
\begin{equation}\label{eq:ceqe}
c_t = y_t
\end{equation}

In (\ref{eq:ceqe}), $y$ is defined as $y_{t+1}= \lambda_{t+1}\,y_{t}$, where ${\lambda_{t+1}}$ is defined as a two state Markov Chain.  Using the properites of Markov process, 
\begin{equation}\label{eq:income1}
y_{t+1}= \prod^t_{i=0}\lambda_{i}\,y_{0}
\end{equation}

Substituting (\ref{eq:ceqe}) into the first order conditions of the household problem, into (\ref{eq:max3}), will obtain an equation that will solve the price system for any asset that will lead to competitive equilibrium.
\begin{equation}
p^0_t(s^t) = \beta^t\,\pi(s^t|s_0)\,\dfrac{u\,'(c^i_t(s^t))}{u\,'(c^i_0(s^0))} \hspace{.75in} \textrm{from (\ref{eq:max3})}
\end{equation}
\begin{equation}\label{eq:ans}
p^0_t(s^t) = \beta^t\,\pi(s^t|s_0)\,\dfrac{u\,'(y_t)}{u\,'(y_0)} \hspace{.75in} \textrm{using (\ref{eq:ceqe})}
\end{equation}
\vspace{.25in}
\\
\item[(c)]  Suppose that one of the consumers markets a derivative asset that promises to pay $.05$ units of consumption each period.  What would the price of the asset be?  
\vspace{.25cm}
\\
From Cochrane, (pg.52 1ed.), using to what he refers to as the \textbf{happy-meal theorem (3.1):}

\begin{equation}\label{eq:happy}
p(x) =  \sum_s \,pc(s)x(s)
\end{equation}

where $x(s)$ denotes an asset's payoff in state of nature $s$, $pc$ is the price of a contingent claim and $(s)$ is used to denote which state $s$ the claim pays off.  From (\ref{eq:happy}), obtain
\vspace{.33cm}
\\
\begin{equation}\label{eq:happy1}
p =  \sum_s \,\beta^t\, \dfrac{u\,'(c_{t+1}))}{u\,'(c_{t})}\,\pi(s^t|s_0)\,x_{t+1}
\end{equation}

In the setup of problem $\#\,1$: $\dfrac{u\,'(c_{t+1}))}{u\,'(c_{t})} = 1$, and $\pi(s^t|s_0) = 1$.  Obtain:

\begin{equation}\label{eq:happy2}
p =  \sum_s \,\beta^t\,x_{t+1}
\end{equation}

Since a derivative asset that promises to pay $.05$ units of consumption each period, 

\begin{displaymath}
p^0_t(s) =  \sum_s \,\beta^t\,(.05) = \dfrac{.05}{1-\beta}
\end{displaymath}


\end{description}

\end{enumerate}

\end{document}