\documentclass[11pt]{article}
\usepackage{amsmath}
\usepackage{lscape}
\textwidth=7.7in  \textheight=10in
\hoffset=-1.5in \voffset=-1.5in
%\pagestyle{empty}
\newcommand{\boxspace}{\parbox{1.42in}}
\newcommand{\hhspace}{\hspace{0.02in}}
%
\newcommand{\bboxspace}{\parbox{2.0in}}
\newcommand{\hhhspace}{\hspace{0.05in}}
\input{tcilatex}
\begin{document}
\begin{enumerate}
\item Consider the standard cake eating problem with a modification of the transition equation for the cake
\begin{equation} \label{eq:trans}
W' = RW -c
\end{equation}

if $R > 1$. the cake yields a positive return, whereas if $R \epsilon (0,1)$ the cake depreciates.
\begin{description} 
\item[(a)] If preferences are of the form $u(c) = \dfrac{c^{1-\sigma}}{1-\sigma}$, calculate the optimal sequence of consumption.  \\What happens when $\sigma=1$? and when $\sigma = 0$?
\vspace{.25cm}
\\
To solve the constrained optimization problem, need to find the sequence of $\{c_t\}_1^T \ \{W_t\}_2^{T+1}$ that satisifies
\begin{equation} \label{eq:maxval1}
V_{t}(W\:') =\textrm{max} \sum_{t=1}^T \: \beta^{(t-1)} \:\: u(c_t)
\end{equation}
subject to the transition equation (\ref{eq:trans}), which holds for $t=1,2,3, \ldots, T.$  Also, subject to nonegative constrants on comsuming the cake given by $c_t \geq 0$ \: and \: $W_{t+1} \geq 0$
\vspace{.15cm}
\\
Adding period 0 consumption, the solution to (\ref{eq:maxval1}) can be rewitten as
\begin{equation} \label{eq:maxval2}
V_{t+1}(W) = \textrm{max}_{\{c_t\}_t^T} \{u(c_t) +  \: \beta V_{t}(W\:') \}
\end{equation}
\vspace{.15cm}
\\
By using the Legrange multipliers, combining the first order conditions we obtain the \textbf{Euler equation} that is necessary condition of optimality for any $t$.
\begin{equation} \label{eq:euler}
u\:'(c_t) = R \, \beta\,  u\,'(c_{t+1})
\end{equation}
\vspace{.15cm}
\\
By letting $u(c) = \dfrac{c^{1-\sigma}}{1-\sigma},  \Rightarrow u\,'(c)= c^{-\sigma}$, subsitute in (\ref{eq:euler}) obtain the following:
\begin{equation} \label{eq:derutil1}
(c_{t})^{-\sigma} = R \, \beta\, (c_{t+1})^{-\sigma}
\end{equation}
\begin{equation} \label{eq:derutil2}
c_{t+1} = (R \, \beta)^{\frac{1}{\sigma}} \, c_{t}
\end{equation}
\vspace{.15cm}
\\
Let $\tilde{\beta} = (R \, \beta)^{\frac{1}{\sigma}}$, and using properites of geometric sequence then (\ref{eq:derutil2}) can be expressed as consumpution of period $T+1$ in terms of period 0. 
\begin{equation} \label{eq:geo1}
c_{t+1} = (\tilde{\beta})^T \, c_{0}
\end{equation}
\vspace{.15cm}
\\
Using the sequence of transition equations from (\ref{eq:trans}), multiply each equation by $R^{T-i}$ where $i=0,1,\ldots, T$.  

\begin{tabular}{c c}
at $t=0$ & \hspace{.5cm} $R^T \, W^1 = R_{T+1}\,W_0- R_T\,c_0$ 
\vspace{.05cm}
\\
at $t=1$ & \hspace{.5cm} $R^{T-1} \, W_2 = R^{T}\,W_1- R^{T-1}\,c_1$ 
\vspace{.05cm}
\\
at $t=2$ & \hspace{.5cm} $R^{T-2} \, W_3 = R^{T-1}\,W_0- R^{T-2}\,c_2$ 
\vspace{.05cm}
\\
$\vdots$ & \hspace{.5cm} $\vdots$ 
\vspace{.05cm}
\\
at $t=T$ & \hspace{.5cm} $R^{0} \, W_{T+1} = R^{0}\,W_T- R^{0}\,c_T$ 
\vspace{.05cm}
\\
\end{tabular}

When sum all the transaction equations obtain the following:

\begin{equation} \label{eq:sumTrans1}
R^0\,W_{T+1} = R^{T+1}\,W_0 - \sum_{t=0}^{T} \: R^{T-t}\,c_{t}
\end{equation}
\vspace{.25cm}
\\
Let consumption at period $W_{T+1}=0$ and intial cake size is known, $W_{0}=\overline{W}_0$.  By subsituting (\ref{eq:geo1}) into (\ref{eq:sumTrans1}) for $c_{i}$, obtain the following expression for $c_0$.
\begin{displaymath} 
R^{T+1}\,\overline{W}_0 = \sum_{t=0}^{T} \: R^{T-t}\,(\tilde{\beta})^t \, c_{0}
\end{displaymath}

\begin{displaymath} 
R^{T+1}\,\overline{W}_0 = R^{T}\,c_{0}\sum_{t=0}^{T} \: R^{-t}\,(\tilde{\beta})^t
\end{displaymath}

\begin{displaymath} 
R\,\overline{W}_0 = c_{0}\sum_{t=0}^{T} \: \left(\dfrac{\tilde{\beta}}{R}\right)^t
\end{displaymath}

\begin{equation} \label{eq:c}
c_{0} = \dfrac{R\,\overline{W}_0} {\sum_{t=0}^{T} \: \left(\dfrac{\tilde{\beta}}{R}\right)^t}
\end{equation}
\vspace{.25cm}
\\
From (\ref{eq:geo1}), subsititing (\ref{eq:c}), obtain a solution for the consumption at any period $t=1,2,\ldots, T$ on the finite horizon.
\begin{equation} \label{eq:cT1}
c_{t} = \dfrac{\tilde{\beta}^{t}\,R\,\overline{W}_0} {\sum_{t=0}^{T} \: \left(\dfrac{\tilde{\beta}}{R}\right)^t}
\end{equation}
\vspace{.25cm}
\\
\textbf{Recall} that $\tilde{\beta} = \left(R \, \beta \right)^{\frac{1}{\sigma}}$.  

\begin{equation} \label{eq:cT2}
c_{t} = \dfrac{\left(R \, \beta \right)^{\frac{t}{\sigma}}\,R\,\overline{W}_0} {\sum_{t=0}^{T} \: \left(\dfrac{\left(R \, \beta \right)^{\frac{1}{\sigma}}}{R}\right)^t}
\end{equation}
\vspace{.25cm}
\\
\textbf{When} $\sigma = 1$ \textbf{obtain:}
\begin{equation} \label{eq:cT3}
c_{t} = \dfrac{\left(R \, \beta \right)^{t}\,R\,\overline{W}_0}{\sum_{t=0}^{T} \: \left(\beta \right)^t}
\end{equation}
\vspace{.25cm}
\\
\textbf{When} $\sigma = 0$ \textbf{obtain:}
\begin{displaymath}
u(c) = c \Rightarrow u\,'(c) = 1
\end{displaymath}
\vspace{.1cm}
\\
From (\ref{eq:euler}), obtain the following:
\begin{equation} \label{eq:cT4}
1 = R\,\beta
\end{equation}

\newpage
\item[(b)] Show that we can reduce the cake eating problem to a system of nonlinear equations of the form $F(W)=0$, the optimal sequence of consumption, $\{c_t\,^{*}\}_{1}^{T}$.  
\vspace{.15cm}
\\
\textbf{Note:}  If find the sequence $\{W_t\,^{*}\}_{1}^{T+1}$, can find sequence of consumption by using (\ref{eq:trans}).
\vspace{.25cm}
\\
Using (\ref{eq:maxval2}), subject to the constaints (\ref{eq:trans}), \textbf{Note:} $c_{t} = RW_{T} - W_{t+1}$, consumption at period $W_{T+2}=0$ and intial cake size is known, $W_{0}=\overline{W}_0$, subsituting in (\ref{eq:euler}), obtain the following system of nonlinear equations with $t$ equations and $t$ unknowns:
\vspace{.25cm}
\\
\begin{displaymath}
F(\mathbf{W}) =
\left[ \begin{array}{c}
u\,'(R\,W_0- W_1) = R\,\beta\,u\,'(R\,W_1- W_2) 
\vspace{.1cm}
\\ 
u\,'(R\,W_1- W_2) = R\,\beta\,u\,'(R\,W_2- W_3)
\vspace{.1cm}
\\
u\,'(R\,W_2- W_3) = R\,\beta\,u\,'(R\,W_3- W_4)
\\ 
\vdots
\\ 
u\,'(R\,W_{T}- W_{T+1}) = R\,\beta\,u\,'(R\,W_{T+1}- W_{T+2})
\vspace{.1cm}
\\ 
\end{array} \right]
\end{displaymath}


\item[(c)]
\end{description}

\newpage

\item Consider the previous version  of the cake eating problem with a law of motion of the form, 
\begin{equation} \label{eq:trans2}
W\,' = y + \delta -c \:, \hspace{.25in} \textrm{where} \hspace{.1in} \delta\,\epsilon\,(0,R)
\end{equation}


\begin{description}
\item[(a)] Formulate the Bellman equation of the infinite horizon problem.
\vspace{.25cm}
\\
As in the case of the previous problem, calculate the optimal sequences, $\{c_t\}_1^{\infty} \ \{W_t\}_2^{\infty}$, that solves the following equation, subject to the constraint in (\ref{eq:trans2}).
\begin{equation} \label{eq:maxv1}
V(W) =\textrm{max} \sum_{t=0}^T \: \beta^{t} \:\: u(c_t)
\end{equation}
\vspace{.25cm}
\\
Equation (\ref{eq:maxv1}) can be rewritten as, subject to (\ref{eq:trans2}), where $0 \leq\, W' \leq W$:
\begin{equation} \label{eq:maxv2}
V(W) =\textrm{max} \{\, u(y+\delta\,W-W') + \beta\, V(W')\, \}
\end{equation}
\vspace{.25cm}

\item[(b)] If preferences are of the form $u(c) = \ln(c)$ and $y=0$, calculate the optimal value function and the implied decision rules.  Does the size of $\delta$ matter for the optimal values?
\vspace{.25cm}
\\
\begin{displaymath}
u(c) = \ln(c) \Rightarrow u\,'(c) = \dfrac{1}{c} \hspace{.5cm} \textrm{and} \hspace{.5cm} y=0 
\end{displaymath}

\begin{equation}\label{eq:trans3}
W\,'= \delta\,W -c \hspace{.25cm} \Rightarrow c = \delta\,W - W'
\end{equation}
\vspace{.25cm}
\\
Using the following form below for the value function, derive the first order conditions to find the optimal policy functions.
\vspace{.1cm}
\\
\begin{equation} \label{eq:guessvalue}
V(W) = \alpha_{0} + \alpha_{1}\,\ln(W) \hspace{.5in} \textrm{with} \Rightarrow V\,'(W)= \dfrac{\alpha_{1}}{W}
\end{equation}

\underline{First Order Conditions (\ref{eq:maxv2})}
\vspace{.25in}
\\
\begin{tabular}{c c c c}
& \hspace{.5in} $\left[W'\right]$ : & \hspace{.25in} $u\,'(c) = \beta\,V\,'(W')$ & \hspace{.5cm}
\vspace{.33cm}
\\
& \hspace{.5in} & \hspace{.25in} $\dfrac{1}{\delta\,W-W'} = \beta \left(\dfrac{\alpha_1}{W'}\right)$ & \hspace{.5cm} \textbf{subsituting} (\ref{eq:trans3}) \textbf{and} (\ref{eq:guessvalue}).  
\vspace{.33cm}
\\
& \hspace{.5in} & \hspace{.25in} $W\,' = \beta\,\alpha_1\,\delta\,W - \alpha_1\,\beta\,W'$ & \hspace{.5cm} 
\end{tabular}
\vspace{.33cm}
\\
Solve for $W'$ to obtain the \textbf{optimal policy function for savings}:
\begin{equation}\label{eq:optpolifunc}
W\,' = \left(\dfrac{\beta\,\alpha_1\delta}{1+\beta\,\alpha_1}\right)W
\end{equation}
\vspace{.5cm}
\\
Subsituting (\ref{eq:optpolifunc})\hspace{.05cm} in (\ref{eq:trans3}) to obtain the \textbf{optimal policy function for consumption}:
\vspace{.25cm}
\\
\begin{displaymath}
c = \delta\,W - \left(\dfrac{\beta\,\alpha_1\delta}{1+\beta\,\alpha_1}\right)W = \left(\dfrac{\delta + \alpha_1\,\delta\,\beta-\alpha_1\,\delta\,\beta}{1+\alpha_1\,\beta}\right)W
\end{displaymath}
\vspace{.25cm}
\\
\begin{equation}\label{eq:optpolifuncC}
c= \left(\dfrac{\delta}{1+\beta\,\alpha_1}\right)W
\end{equation}

\newpage
Need to solve for the coefficents of the value function in (\ref{eq:guessvalue}).  Setting (\ref{eq:guessvalue}) equal to (\ref{eq:maxv2}), and solve for coefficients of (\ref{eq:guessvalue}) $\{\alpha_0, \,\alpha_1\}$.  This will be done by subsituting (\ref{eq:optpolifuncC}) \hspace{.05cm} and (\ref{eq:optpolifunc}) into (\ref{eq:maxv2}).
\\
Solving for $\{\alpha_0, \,\alpha_1\}$ and subsituting into (\ref{eq:optpolifuncC}) \hspace{.05cm} and (\ref{eq:optpolifunc}) will obtain the optimal value function.
\vspace{.25in}
\\
\begin{tabular}{c c c}
& \hspace{1in} $\alpha_{0} + \alpha_{1}\,\ln(W) = u(c) + \beta\,^t\,V(W\,')$ & \hspace{.25in} \textbf{from} (\ref{eq:guessvalue}) \textbf{and} (\ref{eq:maxv2})
\vspace{.25in}
\\
& \hspace{.75in} $\alpha_{0} + \alpha_{1}\,\ln(W) = \ln(c) + \beta\,(\alpha_{0} + \alpha_{1}\,\ln(W\,'))$ & \hspace{.25in} \textbf{from given} $u(c) = \ln(c)$ \textbf{and} (\ref{eq:guessvalue})
\end{tabular}
\vspace{.25in}
\\
\begin{tabular}{c c c}
& \hspace{.5in} $\alpha_{0} + \alpha_{1}\,\ln(W) = \ln\left(\dfrac{\delta\,W}{1+\beta\,\alpha_1}\right) + \beta\,\alpha_{0} + \beta\,\alpha_{1}\,\ln\left(\dfrac{\beta\,\alpha_1\delta\,W}{1+\beta\,\alpha_1}\right)$ & \hspace{.25in} \textbf{from} (\ref{eq:optpolifuncC}) \textbf{and} (\ref{eq:optpolifunc})
\end{tabular}
\vspace{.25in}
\\
\begin{displaymath}
\alpha_{0} + \alpha_{1}\,\ln(W) = \underbrace{\ln\left(\dfrac{\delta}{1+\beta\,\alpha_1}\right) + \beta\,\alpha_0 + \beta\,\alpha_{1}\,\ln\left(\dfrac{\beta\,\alpha_1\delta}{1+\beta\,\alpha_1}\right)}_{constant \,=\,\alpha_0} + \underbrace{(1+\beta\,\alpha_{1})}_{\alpha_1}\ln(W)
\end{displaymath}
\vspace{.25in}
\\
Therefore, able to setup the following equations and solve for $\{\alpha_0, \,\alpha_1\}$.
\begin{displaymath}
\alpha_1 = 1 + \beta\,\alpha_1
\end{displaymath}

\begin{equation}\label{eq:alpha1}
\alpha_1 = \dfrac{1}{1-\beta}
\end{equation}
\vspace{.25in}
\\
\begin{displaymath}
\alpha_0 = \ln\left(\dfrac{\delta}{1+\beta\,\alpha_1}\right) + \beta\,\alpha_0 + \beta\,\alpha_{1}\,\ln\left(\dfrac{\beta\,\alpha_1\delta}{1+\beta\,\alpha_1}\right)
\end{displaymath}
\vspace{.25in}
\\
\begin{tabular}{c c c}
& \hspace{.25in} $\alpha_0 = \ln\left(\dfrac{\delta}{1+\dfrac{\beta}{1-\beta}}\right) + \beta\,\alpha_0 + \dfrac{\beta}{1-\beta}\,\ln\left(\dfrac{\dfrac{\beta\,\delta}{1-\beta}}{1+\dfrac{\beta}{1-\beta}}\right)$ & \hspace{.25in} \textbf{Subsituting} (\ref{eq:alpha1})
\end{tabular}
\vspace{.25in}
\\
\begin{tabular}{c c c}
& \hspace{1.25in} $\alpha_0(1-\beta) = \ln\left(\delta\,(1-\beta)\right) + \dfrac{\beta}{1-\beta}\,\ln(\beta\,\delta)$
\end{tabular}
\vspace{.25in}
\\
\begin{equation}\label{eq:alpha0}
\alpha_0  = \dfrac{1}{1-\beta}\ln\left(\delta\,(1-\beta)\right) + \dfrac{\beta}{(1-\beta)^2}\,\ln(\beta\,\delta)
\end{equation}
\vspace{.15in}
\\

\newpage

Using (\ref{eq:alpha0}) and (\ref{eq:alpha1}), able to obtain the \textbf{optimal policy function} and the \textbf{optimal value function}
\vspace{.25in}
\\
\begin{tabular}{c c c}
& \hspace{1.25in} $c = \left(\dfrac{\delta}{1+\alpha_1\,\beta}\right)W$ & \hspace{.15in} \textbf{from} (\ref{eq:optpolifuncC})
\vspace{.25in}
\\
& \hspace{1.25in} $c = \left(\dfrac{\delta}{1+\dfrac{\beta}{1-\beta}}\right)W$ & \hspace{.25in} \textbf{subsituting} (\ref{eq:alpha1})
\end{tabular}
\vspace{.25in}
\\
\begin{equation}\label{eq:optC}
c  = \delta(1-\beta)W
\end{equation}
\vspace{.25in}
\\
\begin{tabular}{c c c}
& \hspace{1.25in} $W\,' = \delta\,W-C$ & \hspace{.15in} \textbf{from} (\ref{eq:trans3})
\end{tabular}
\vspace{.25in}
\\
\begin{equation}\label{eq:optW}
W\,' = \delta\,\beta\,W
\end{equation}
\vspace{.25in}
\\
Therefore, (\ref{eq:optW}) and (\ref{eq:optC}) is the \textbf{optimal policy function}.  Using the solutions for $\{\alpha_0, \,\alpha_1\}$ from (\ref{eq:alpha1}) and (\ref{eq:alpha0}) able to obtain the following \textbf{optimal value function}
\vspace{.25in}
\\
\begin{equation}\label{eq:opvalue}
V(w) = \dfrac{1}{1-\beta}\ln\left(\delta\,(1-\beta)\right) + \dfrac{\beta}{(1-\beta)^2}\,\ln(\beta\,\delta) + \dfrac{1}{1-\beta}\ln(W)
\end{equation}

Does

\item[(c)]

\end{description}

\newpage

\item Consider the instanteous utility function that depends on previous consumption $u(c_{t-1}, c_t)$.  Previous consumption affects present consumption, so consumers have to take into account that today's consumption choice will also affect future utility besides the size of the cake.

\begin{description}
\item[(a.)]  Solve the $T$-period cake eating problem using this preferences.  
\vspace{.33cm}
\\
The value function, using the conditions where the previous consumption affects present consumption, 


Calculate the optimal sequence, $\{c_t\}_0^{\infty}$, that solves the following equation, subject to the constraint listed.
\begin{eqnarray} \label{eq:maxv3}
\textrm{max} \sum_{t=0}^T \: \beta^{t} \:\: u(c_{t-1},c_t) 
\vspace{.25cm}
\\
\textrm{subject to} \hspace{.25in} \sum_{t=0}^T c_t + W_{T+1} = W_0 \label{eq:trans4}
\end{eqnarray}
\vspace{.25cm}



\end{description}



\end{enumerate}

\end{document}