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%	PACKAGES AND THEMES
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% ----------------------------------------------------------------------------------------
%	TITLE PAGE
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\title[]{Macro II} % The short title appears at the bottom of every slide, the full
% title is only on the title page

\author{Professor Griffy} % Your name
\institute[University at Albany, SUNY] % Your institution as it will appear on the bottom of
% every slide, may be shorthand to save space
{
UAlbany  \ % Your institution for the title page
}
\date{Spring 2023} % Date, can be changed to a custom date

\begin{document}

\begin{frame}
  \titlepage % Print the title page as the first slide
\end{frame}




% ----------------------------------------------------------------------------------------
%	PRESENTATION SLIDES
% ----------------------------------------------------------------------------------------

% ------------------------------------------------
\section{Introduction} % Sections can be created in order to organize your presentation into discrete blocks, all sections and subsections are automatically printed in the table of contents as an overview of the talk
% ------------------------------------------------

\begin{frame}
  \frametitle{Introduction}
  \begin{itemize}
  \item Today: Market structure
  \item Complete markets:
    \begin{itemize}
    \item Arrow-Debreu structure (time-0 contingent claims);
    \item Arrow securities (sequentially traded one-period claims).
    \end{itemize}
  \item Homework 3 due Thursday.
  \item No class Thursday (3/2).
  \item Midterm Thursday after Spring Break (3/23).
  \end{itemize}
\end{frame}

% ------------------------------------------------
\section{Complete Markets}
% ------------------------------------------------


\begin{frame}\frametitle{Complete markets}
\begin{itemize}
\item Individuals in the economy have access to a comprehensive set of risk-sharing contracts:
  \begin{itemize}
  \item They can contract to insure against any event or sequence of events.
  \item They write these contracts with other agents in the economy.
  \end{itemize}
\item Will lead to
  \begin{itemize}
  \item Perfect risk sharing
  \item i.e., representative agent.
  \end{itemize}
\end{itemize}
\end{frame}

% ------------------------------------------------

\begin{frame}\frametitle{Complete markets}
  \begin{itemize}
  \item Define unconditional probability of sequence of shocks $s^{t} = [s_{0},s_{1},...,s_{t}]$ to be $\pi_{t}(s^{t})$.
  \item Assume there are $i = 1,...,I$ consumers, each of whom receives a stochastic endowment $y_{t}^{i}(s^{t})$.
  \item They purchase a consumption plan that stipulates consumption for any history of shocks and yields:
    \begin{align*}
      U_{i}(c^{i}) &= \sum_{t = 0}^{\infty}\sum_{s^{t}}\beta^{t}u_{i}[c_{t}^{i}(s^{t})]\pi_{t}(s^{t})
    \end{align*}
  \item These contracts yield expected lifetime utility, where $\lim_{s\rightarrow 0}u_{i}'(c) = +\infty$
  \end{itemize}
\end{frame}

% ------------------------------------------------

\begin{frame}\frametitle{Complete markets}
  \begin{itemize}
  \item They purchase a consumption plan that stipulates consumption for any history of shocks and yields:
    \begin{align*}
      U_{i}(c^{i}) &= \sum_{t = 0}^{\infty}\sum_{s^{t}}\beta^{t}u_{i}[c_{t}^{i}(s^{t})]\pi_{t}(s^{t})
    \end{align*}
  \item And are subject to a feasibility constraint:
    \begin{align*}
      \sum_{i}c_{t}^{i}(s^{t}) \leq \sum_{i}y_{t}^{i}(s^{t})\;\forall\;t,\;s^{t}
    \end{align*}
  \item These contracts determine how to split resources at each $t$.
  \item i.e., they insure individuals ex-ante against income risk.
  \end{itemize}
\end{frame}

% ------------------------------------------------

\begin{frame}\frametitle{Contingent claims trading structure}
  \begin{itemize}
  \item Arrow-Debreu structure: contract at time $t = 0$ on \textit{every} possible sequence of shocks.
  \end{itemize}
    \centering\includegraphics[width=0.6\textwidth]{./ArrowDebreuStructure.png}
    \begin{itemize}
    \item Each node represents a possible sequence of shocks.
    \item A consumption plan would specify consumption at each node at each time.
    \end{itemize}
\end{frame}

% ------------------------------------------------

\begin{frame}\frametitle{Sequential trading structure}
  \begin{itemize}
  \item Arrow securities: re-contract at ever $t$ given the history of shocks $s^{t}$.
  \end{itemize}
    \centering\includegraphics[width=0.6\textwidth]{./ArrowSecurities.png}
    \begin{itemize}
    \item At $t = 2$, contract for two possible realizations.
    \end{itemize}
\end{frame}

% ------------------------------------------------

\begin{frame}\frametitle{Trading structure}
  \begin{itemize}
  \item Arrow-Debreu structure: contract at time $t = 0$ on \textit{every} possible sequence of shocks.
  \item Arrow securities: re-contract at ever $t$ given the history of shocks $s^{t}$.
  \item Do these trading structure yield the same equilibrium allocation? Yes.
  \item Important property:
    \begin{itemize}
    \item Under either structure, allocations are a function of the aggregate state only (\& initial conditions).
    \item i.e., allocation depends only on $\sum_{i = 1}^{I}y_{t}^{i}(s^{t})$
    \end{itemize}
  \item Leads to representative agent structure.
  \end{itemize}
\end{frame}

% ------------------------------------------------
\section{Planner's Allocation}
% ------------------------------------------------


\begin{frame}\frametitle{Planner's Problem}
\begin{itemize}
\item First, we will find the Pareto optimal allocation.
\item i.e., the allocation from solving the Social Planner's problem:
  \begin{align*}
    \max_{c^{i}} W = \sum_{i = 1}^{I}\lambda_{i}U_{i}(c^{i})
  \end{align*}
\item where $\lambda_{i}$ is a ``Pareto weight,'' i.e., how much Planner values individual $i$ relative to others.
\item Constrained maximization:
  \begin{align*}
    L = \sum_{t = 0}^{\infty}\sum_{s^{t}}\{\sum_{i = 1}^{I}\lambda_{i}\beta^{t}u_{i}(c_{t}^{i})\pi_{t}(s^{t}) + \theta_{t}(s^{t})\sum_{i = 1}^{I}[y_{t}^{i}(s^{t}) - c_{t}^{i}(s^{t})]\}
  \end{align*}
\item i.e., maximize weighted expected utility subject to the feasibility constraint (multiplier $\theta$)
\end{itemize}
\end{frame}

% ------------------------------------------------

\begin{frame}\frametitle{Planner's Problem}
\begin{itemize}
\item Constrained maximization:
  \begin{align*}
    L = \sum_{t = 0}^{\infty}\sum_{s^{t}}\{\sum_{i = 1}^{I}\lambda_{i}\beta^{t}u_{i}(c_{t}^{i})\pi_{t}(s^{t}) + \theta_{t}(s^{t})\sum_{i = 1}^{I}[y_{t}^{i}(s^{t}) - c_{t}^{i}(s^{t})]\}
  \end{align*}
\item FOC in $c_{t}^{i}$:
  \begin{align*}
    \beta^{t} u_{i}'(c_{t}^{i}(s^{t}))\pi_{t}(s^{t}) = \lambda_{i}^{-1}\theta_{t}(s^{t})
  \end{align*}
\item How is this allocated across consumers?
  \begin{align*}
    \frac{u_{i}'(c_{t}^{i}(s^{t}))}{u_{1}'(c_{t}^{1}(s^{t}))} &= \frac{\lambda_{1}}{\lambda_{i}}\\
    \rightarrow c_{t}^{i}(s^{t}) &= u_{i}'^{-1}(\lambda_{i}^{-1}\lambda_{1}u_{1}'(c_{t}^{1}(s^{t})))
  \end{align*}
\item Often, assume $\lambda_{i} = \lambda_{1}\forall\;i\rightarrow c_{t}^{i}(s^{t}) = u_{i}'^{-1}(u_{1}'(c_{t}^{1}(s^{t})))$
\end{itemize}
\end{frame}

% ------------------------------------------------

\begin{frame}\frametitle{Planner's Problem}
\begin{itemize}
\item Allocation:
  \begin{align*}
    c_{t}^{i}(s^{t}) &= u_{i}'^{-1}(\lambda_{i}^{-1}\lambda_{1}u_{1}'(c_{t}^{1}(s^{t})))
  \end{align*}
\item Sub into resource constraint:
  \begin{align*}
    \sum_{i}u_{i}'^{-1}(\lambda_{i}^{-1}\lambda_{1}u_{1}'(c_{t}^{1}(s^{t}))) = \sum_{i}y_{t}^{i}(s^{t})
  \end{align*}
\item i.e., the resource allocation depends only on aggregate endowment and weights of each consumer.
\end{itemize}
\end{frame}

% ------------------------------------------------
\section{Time $0$ trading}
% ------------------------------------------------

\begin{frame}\frametitle{Decentralized allocations}
\begin{itemize}
\item We know that the optimal allocation is given by
  \begin{align*}
    \sum_{i}u_{i}'^{-1}(\lambda_{i}^{-1}\lambda_{1}u_{1}'(c_{t}^{1}(s^{t}))) = \sum_{i}y_{t}^{i}(s^{t})
  \end{align*}
\item Can we achieve the same allocation under different trading regimes?
\item Specifically, does the decentralized economy achieve the same allocation?
\end{itemize}
\end{frame}

% ------------------------------------------------

\begin{frame}\frametitle{Consumer's problem}
\begin{itemize}
\item Consumer's problem: maximize
    \begin{align*}
      U_{i}(c^{i}) &= \sum_{t = 0}^{\infty}\sum_{s^{t}}\beta^{t}u_{i}[c_{t}^{i}(s^{t})]\pi_{t}(s^{t})
    \end{align*}
  \item subject to
    \begin{align*}
      \sum_{t=0}^{\infty}\sum_{s^{t}}q_{t}^{0}(s^{t})c_{t}^{i}(s^{t}) \leq \sum_{t=0}^{\infty}\sum_{s^{t}}q_{t}^{0}(s^{t})y_{t}^{i}(s^{t})
    \end{align*}
\end{itemize}
\end{frame}

% ------------------------------------------------

\begin{frame}\frametitle{Consumer's problem}
\begin{itemize}
\item Yields the following:
  \begin{align*}
    \beta^{t}u_{i}'[c_{t}^{i}(s^{t})]\pi_{t}(s^{t}) &= \mu_{i}q_{t}^{0}(s^{t})\\
    \frac{u_{i}'(c_{t}^{i}(s^{t}))}{u_{1}'(c_{t}^{1}(s^{t}))} &= \frac{\mu_{i}}{\mu_{1}}
  \end{align*}
\item which implies
  \begin{align*}
        \sum_{i}u_{i}'^{-1}(\mu_{1}^{-1}\mu_{i}u_{1}'(c_{t}^{1}(s^{t}))) &= \sum_{i}y_{t}^{i}(s^{t})
  \end{align*}
\end{itemize}
\end{frame}

% ------------------------------------------------

\begin{frame}
	\frametitle{Competitive Equilibrium}
	{\bf Definition} \	A competitive equilibrium is a price system $\{q^0_t(s^t)\}_{t=0}^\infty$ and allocation $\{c^{i*}\}_{i \in \mathcal{I}}$ such that
		\begin{enumerate}
			\item Given a price system, each individaul $i \in \mathcal{I}$ solves the following problem:
				\begin{align*}
					\{c^{i*}_t(s^t)\}_{t=0}^\infty = \ & arg \max_{\{c^i_t(s^t)\}_{t=0}^\infty} \ \sum_{t=0}^\infty \sum_{s^t} \beta^t  u\Big(c^i_t(s^t)\Big) \pi_t(s^t) \\
					& s.t. \hs \sum_{t=0}^\infty \sum_{s^t} q_t^0(s^t) c_t^i(s^t) \leq \sum_{t=0}^\infty \sum_{s^t} q_t^0(s^t) y_t^i(s^t)
				\end{align*}
			\item On every history $s^t$ at time $t$, market clears
				\[ \sum_{i \in \mathcal{I}} c_t^{i*}(s^t) = \sum_{i \in \mathcal{I}} y_t^i(s^t)  \]
		\end{enumerate}
Rules out economies with externalities, incomplete markets, etc.
\end{frame}

% ------------------------------------------------

\begin{frame}\frametitle{First Welfare Theorem}
\begin{itemize}
\item First welfare theorem:\\
\textit{Let $c$ be a competitive equilibrium allocation. Then $c$ is pareto efficient.}
\item Equivalence: Competitive equilibrium is a specific Pareto optimal allocation in which $\lambda_{i} = \mu_{i}^{-1}$.
\end{itemize}
\end{frame}

% ------------------------------------------------
\section{Sequential Trading}
% ------------------------------------------------

\begin{frame}\frametitle{Sequential trading}
\begin{itemize}
\item Now, we will consider an economy with sequential trades.
\item i.e., each period agents meet and trade state-contingent bonds
\item Recall from asset pricing:
\begin{equation*}
p_{t}=\beta E_{t}\left( \frac{u^{' }\left( d_{t+1}\right) }{u^{'
}\left( d_{t}\right) }\left( p_{t+1}+d_{t+1}\right) \right)
\end{equation*}%
\item where the expectation is over realizations of $s_{t + 1}$, which determines $d_{t + 1}$.
\item Price is determined by payout of asset across all different realizations.
\item i.e., asset that provides good return across all realizations: expensive.
\end{itemize}
\end{frame}

% ------------------------------------------------

\begin{frame}\frametitle{Market clearing}
\begin{itemize}
\item Recall from asset pricing that the net bond position of the economy equaled zero.
\item i.e., $\sum_{i}b_{t+1}^{i} = 0$.
\item Same in this context.
\item Some are borrowing and some are saving (in principle, if there were heterogeneity).
\item This must net to zero.
\end{itemize}
\end{frame}

% ------------------------------------------------

\begin{frame}\frametitle{Restriction: No Ponzi Schemes}
\begin{itemize}
\item Must ensure that agents never take out too much debt.
\item Natural debt limit:
  \begin{align*}
    A_{t}^{i}(s^{t}) = \sum_{\tau = t}^{\infty}\sum_{s^{\tau}|s^{t}}q_{\tau}^{t}(s^{\tau})y_{\tau}^{i}(s^{\tau})
  \end{align*}
\item This is the amount that the agent could borrow and still commit to repay.
\item Rules out Ponzi schemes.
\end{itemize}
\end{frame}

% ------------------------------------------------

\begin{frame}\frametitle{Sequential problem}
\begin{itemize}
\item Consumer's problem: maximize
    \begin{align*}
      U_{i}(c^{i}) &= \sum_{t = 0}^{\infty}\sum_{s^{t}}\beta^{t}u_{i}[c_{t}^{i}(s^{t})]\pi_{t}(s^{t})
    \end{align*}
  \item subject to
    \begin{align*}
      c_{t}^{i} + \sum_{s^{t+1}}Q_{t}(s_{t + 1}|s^{t})a_{t+1}^{i}(s_{t + 1},s^{t}) &\leq y_{t}^{i}(s^{t}) + a_{t}^{i}(s^{t})\\
      -a_{t + 1}^{i}(s^{t + 1}) &\geq -A_{t + 1}^{i}(s^{t + 1})
    \end{align*}
  \item where $Q_{t}$ is a pricing kernel: price of one unit of consumption given realization $s_{t + 1}$ and history $s^{t}$.
\end{itemize}
\end{frame}

% ------------------------------------------------

\begin{frame}\frametitle{Sequential allocation}
\begin{itemize}
\item Solving the previous problem yields the following Euler Equation:
\begin{equation*}
Q_{t}(s_{t+1}|s^{t})=\beta \left( \frac{u^{' }\left( c_{t+1}^{i}(s^{t + 1})\right) }{u^{'
}\left( c_{t}^{i}(s^{t})\right) }\pi_{t}(s^{t + 1}|s^{t}) \right)
\end{equation*}%
\item Same as the asset pricing specification from earlier.
\item Taking the expectation of this expression across all possible realizations of $s^{t + 1}$ yields the price, $Q$.
\end{itemize}
\end{frame}

% ------------------------------------------------

\begin{frame}
	\frametitle{Sequential Trading - Competitive Equilibrium}
	\footnotesize
	{\bf Definition} \	A competitive equilibrium is a price system $\big\{ \{Q_t(s_{t+1}|s^t)\}_{s_{t+1} \in S}\big\}_{t=0}^\infty$, an allocation $\Big\{ \big\{\tilde c^i_t(s^t), \ \{\tilde a^i_{t+1}(s_{t+1}, s^t)\}_{s_{t+1}\in S}\big\}_{t=0}^\infty \Big\}_{i \in \mathcal{I}}$, an initial distribution of wealth $\{a^i_0(s_0) = 0 \}_{i \in \mathcal{I}}$, and a collection of natural borrowing limits $\Big\{ \big\{ \{A^i_{t+1}(s_{t+1}, s^t)\}_{s_{t+1}\in S}\big\}_{t=0}^\infty \Big\}_{i \in \mathcal{I}}$ such that
	\begin{enumerate}
		\item Given a price system, an initial distribution of wealth, and a collection of natural borrowing limits, each individual $i \in \mathcal{I}$ solves the workers problem.
		\item On every history $s^t$ at time $t$, markets clear.
			\begin{align*}
				\sum_{i \in \mathcal{I}} c_t^{i}(s^t) &= \sum_{i \in \mathcal{I}} y_t^i(s^t) \tag{Commodity market clearing} \\
				\sum_{i \in \mathcal{I}} a_{t+1}^{i}(s_{t+1}, s^t) &= 0 \hs \forall \ s_{t+1} \in S \tag{Asset market clearing} \\
			\end{align*}
	\end{enumerate}
\end{frame}

% --------------------------------------------------------------


% \begin{frame}\frametitle{Equilibrium definition}
% \begin{itemize}
% \item A competitive equilibirum with sequential trading of one-period state-contingent bonds is an initial distribution, natural borrowing limits for all $t$, a feasible allocation of consumption, and a pricing kernel such that
%   \begin{enumerate}
%   \item the consumption and portfolio allocation solve the consumer's problem.
%   \item allocations and portfolios satisfy feasibility.
%   \end{enumerate}
% \end{itemize}
% \end{frame}

% % ------------------------------------------------

\begin{frame}\frametitle{Equivalence of allocations}\label{mainequivalence}
\begin{itemize}
\item Is this allocation also a time-$0$ trading allocation?
\item Yes. Suppose that the pricing kernel takes the following form
\begin{align*}
q_{t + 1}^{0}(s^{t + 1}) &= Q_{t}(s_{t + 1}|s^{t})q_{t}^{0}(s^{t})\\
\frac{q_{t + 1}^{0}(s^{t + 1})}{q_{t}^{0}(s^{t})} &= Q_{t}(s_{t + 1}|s^{t})
\end{align*}%
\item That is, the price of 1 unit of consumption in period $t + 1$ is the same regardless of whether you purchased that consumption last period or in period 0.
\item When this holds, sequential allocation coincides with time-$0$ trading allocation, subject to initial distribution.
\item Formal proof (check on your own): \hyperlink{equivalence}{\beamergotobutton{formal proof}}
\end{itemize}
\end{frame}

% ------------------------------------------------

\begin{frame}
  \frametitle{Conclusion}
  \begin{itemize}
  \item Midterm Thursday after break (3/21).
  \item HW4 between now and then.
  \end{itemize}
\end{frame}

% ------------------------------------------------

\appendix
\section{Additional Material}

% ------------------------------------------------



\begin{frame}\label{equivalence}
	\frametitle{Equivalence of allocations}
		\begin{align*}
			Q_t(s_{t+1}|s^t) &= \frac{q^0_{t+1}(s^{t+1})}{q^0_{t}(s^{t})} \hs \Rightarrow \hs \beta \frac{ u'\Big( \tilde c^{i}_{t+1}(s^{t+1}) \Big) }{u'\Big( \tilde c^{i}_t(s^t) \Big)} \pi_t(s^{t+1}|s^t)  \nonumber\\&= \beta \frac{ u'\Big( c^{i*}_{t+1}(s^{t+1}) \Big) }{u'\Big( c^{i*}_t(s^t) \Big)} \pi_t(s^{t+1}|s^t)
		\end{align*}
\hyperlink{mainequivalence}{\beamerreturnbutton{back}}
\end{frame}

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\begin{frame}
	\frametitle{Guess for portfolio}
		On every history $s^t$ at time $t$,
			\begin{align*}
				\tilde a ^i_{t+1}(s_{t+1}, s^t) = \sum_{\tau = t+1}^{\infty} \ \sum_{s^\tau | (s_{t+1}, s^{t})} \frac{q^0_{\tau}(s^{\tau})}{q^0_{t+1}(s^{t+1})} \Big( c_{\tau}^{i*}(s^\tau) - y_\tau^i(s^\tau) \Big) \hs \forall \ s_{t+1} \in S
			\end{align*}
		Value of this portfolio expressed in terms of the date $t$, history $s^t$ consumption good is $\sum_{s_{t+1} \in S} \tilde a ^i_{t+1}(s_{t+1}, s^t) Q_t(s_{t+1} | s^t) = $
			\begin{align*}
				 & = \sum_{s_{t+1} \in S} \ \sum_{\tau = t+1}^{\infty} \ \sum_{s^\tau | (s_{t+1}, s^{t})} \frac{q^0_{\tau}(s^{\tau})}{q^0_{t+1}(s^{t+1})} \Big( c_{\tau}^{i*}(s^\tau) - y_t^i(s^\tau) \Big) Q_t(s_{t+1} | s^t) \\
				& = \sum_{s_{t+1} \in S} \ \sum_{\tau = t+1}^{\infty} \ \sum_{s^\tau | (s_{t+1}, s^{t})} \frac{q^0_{\tau}(s^{\tau})}{\cancel{q^0_{t+1}(s^{t+1})}} \Big( c_{\tau}^{i*}(s^\tau) - y_t^i(s^\tau) \Big) \frac{\cancel{q^0_{t+1}(s^{t+1})}}{q^0_{t}(s^{t})} \\
				& = \sum_{\tau = t+1}^{\infty} \ \sum_{s^\tau | s^{t} } \frac{q^0_{\tau}(s^{\tau})}{q^0_{t}(s^{t})} \Big( c_{\tau}^{i*}(s^\tau) - y_t^i(s^\tau) \Big)
			\end{align*}
\hyperlink{mainequivalence}{\beamerreturnbutton{back}}
\end{frame}

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\begin{frame}
	\frametitle{Verify portfolio}
	\footnotesize
	On history $s^0 = s_0$ at time $t=0$, assume that $a^i_0(s_0) = 0$. Then
	\begin{align*}
		\tilde{c}^i_0(s_0) + \sum_{s_{1} \in S} \tilde a ^i_{1}(s_{1}, s_0) Q_1(s_{1} | s_0) &= y^i_0(s_0) + 0 \\
		\tilde{c}^i_0(s_0) + \sum_{\tau = 1}^{\infty} \ \sum_{s^\tau | s_{0} } \frac{q^0_{\tau}(s^{\tau})}{q^0_{0}(s_{0})} \Big( c_{\tau}^{i*}(s^\tau) - y_t^i(s^\tau) \Big)  &= y^i_0(s_0) + 0 \\
		q^0_{0}(s_{0}) c^{i*}_0(s_0) + \sum_{\tau = 1}^{\infty} \ \sum_{s^\tau | s_{0} } q^0_{\tau}(s^{\tau}) \Big( c_{\tau}^{i*}(s^\tau) - y_t^i(s^\tau) \Big)  &= q^0_{0}(s_{0}) y^i_0(s_0) \tag{\ if $\tilde{c}^i_0(s_0) =c^{i*}_0(s_0)$ \ } \\
		\sum_{t=0}^\infty \sum_{s^t} q_t^0(s^t) y_t^i(s^t) &=  \sum_{t=0}^\infty \sum_{s^t} q_t^0(s^t) c_t^{i*}(s^t)
	\end{align*}
	Therefore, given $\tilde c^i_0(s_0) = c^{i*}_0(s_0)$, portfolio $\{\tilde a^i_{1}(s_{1}, s_0)\}_{s_{1} \in S}$ is affordable. \\
\hyperlink{mainequivalence}{\beamerreturnbutton{back}}
\end{frame}

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\begin{frame}
	\frametitle{Verify portfolio}
	\footnotesize
	On history $s^t$ at time $t$, assume that $\tilde a^i_t(s^t) = \sum_{\tau = t}^{\infty} \ \sum_{s^\tau | s^{t}} \frac{q^0_{\tau}(s^{\tau})}{q^0_{t}(s^{t})} \Big( c_{\tau}^{i*}(s^\tau) - y_\tau^i(s^\tau) \Big)$. Then
	\begin{align*}
	\tilde{c}^i_t(s^t) &+ \sum_{s_{t+1} \in S} \tilde a ^i_{t+1}(s_{t+1}, s^t) Q_t(s_{t+1} | s^t)  = y^i_t(s^t) \\&+ \sum_{\tau = t}^{\infty} \ \sum_{s^\tau | s^{t}} \frac{q^0_{\tau}(s^{\tau})}{q^0_{t}(s^{t})} \Big( c_{\tau}^{i*}(s^\tau) - y_\tau^i(s^\tau) \Big)
\end{align*}
\begin{align*}
	\tilde{c}^i_t(s^t) &+ \sum_{\tau = t+1}^{\infty} \ \sum_{s^\tau | s^{t} } \frac{q^0_{\tau}(s^{\tau})}{q^0_{t}(s^{t})} \Big( c_{\tau}^{i*}(s^\tau) - y_t^i(s^\tau) \Big)   = y^i_t(s^t) \\&+ \sum_{\tau = t}^{\infty} \ \sum_{s^\tau | s^{t}} \frac{q^0_{\tau}(s^{\tau})}{q^0_{t}(s^{t})} \Big( c_{\tau}^{i*}(s^\tau) - y_\tau^i(s^\tau) \Big)
\end{align*}
\hyperlink{mainequivalence}{\beamerreturnbutton{back}}
\end{frame}

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\begin{frame}
	\frametitle{Verify portfolio}
	\footnotesize
	On history $s^t$ at time $t$, assume that $\tilde a^i_t(s^t) = \sum_{\tau = t}^{\infty} \ \sum_{s^\tau | s^{t}} \frac{q^0_{\tau}(s^{\tau})}{q^0_{t}(s^{t})} \Big( c_{\tau}^{i*}(s^\tau) - y_\tau^i(s^\tau) \Big)$. Then

\begin{align*}
	q^0_{t}(s^{t}) c^{i*}_t(s^t) &+ \sum_{\tau = t+1}^{\infty} \ \sum_{s^\tau | s^{t} } q^0_{\tau}(s^{\tau}) \Big( c_{\tau}^{i*}(s^\tau) - y_t^i(s^\tau) \Big)   = q^0_{t}(s^{t})y^i_t(s^t) \\&+ \sum_{\tau = t}^{\infty} \sum_{s^\tau | s^{t}} q^0_{\tau}(s^{\tau}) \Big( c_{\tau}^{i*}(s^\tau) - y_\tau^i(s^\tau) \Big) \tag{\ if $\tilde{c}^i_t(s^t) =c^{i*}_t(s^t)$ \ }\\
	&\rightarrow\sum_{\tau = t}^{\infty} \ \sum_{s^\tau | s^{t}} q^0_{\tau}(s^{\tau}) \Big( c_{\tau}^{i*}(s^\tau) - y_\tau^i(s^\tau) \Big) = \sum_{\tau = t}^{\infty} \ \sum_{s^\tau | s^{t}} q^0_{\tau}(s^{\tau}) \Big( c_{\tau}^{i*}(s^\tau) - y_\tau^i(s^\tau) \Big)
	\end{align*}
	Therefore, given $\tilde c^i_t(s^t) = c^{i*}_t(s^t)$, portfolio $\{\tilde a^i_{t+1}(s_{t+1}, s^t)\}_{s_{t+1} \in S}$ is affordable. \\
\hyperlink{mainequivalence}{\beamerreturnbutton{back}}
\end{frame}

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\end{document}
