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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 2021} % 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: Financial frictions
\item Introduce 2-period model that captures key insights about financial frictions:
  \begin{itemize}
  \item uncertain returns on financial assets;
  \item caused by information frictions;
  \item can have real effects on economy.
  \end{itemize}
\item Will be a new homework soon (probably 2 more total).
\end{itemize}
\end{frame}

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

\section{Motivation}

\begin{frame}\frametitle{Why study financial frictions?}
\begin{itemize}
\item Data show a strong correlation between financial conditions and real
economic activity

\item What does this mean?

\begin{itemize}
\item Real activity affects financial conditions

\item Financial dynamics amplify or extend the effects of real shocks: the
"financial accelerator"

\item Financial shocks affect the real economy
\end{itemize}
\item In an Arrow-Debreu world, financial frictions do not exist. We need

\begin{itemize}
\item Incomplete markets to create frictions

\item Heterogeneity to make frictions relevant
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}\frametitle{What we need}
\begin{itemize}

\item Incompleteness

\begin{itemize}
\item Can be imposed exogenously

\item Endogenous incompleteness

\begin{itemize}
\item Incomplete information

\item Limited enforcement (of repayment)
\end{itemize}
\end{itemize}

\item Restricting internal financing

\begin{itemize}
\item Finite life spans

\item Heterogeneous discounting

\item Tax incentives
\end{itemize}
\end{itemize}
\end{frame}

\section{Simplified Model}

\begin{frame}\frametitle{A two-period model}
\begin{itemize}
\item Two periods, 1 and 2: use primes to denote period-2 values

\begin{itemize}
\item Period 1: Produce using capital and labor, invest in future capital

\item Period 2: Produce using capital accumulated in previous period
\end{itemize}

\item Two types of agents with risk-neutrality (exogenously given)

\begin{itemize}
\item Unit mass of workers who maximize%
\begin{equation*}
E\left( c-\frac{\ell ^{2}}{2}+\beta c^{\prime }\right) .
\end{equation*}

\item Unit mass of entrepreneurs who maximize%
\begin{equation*}
E(c+\beta c^{\prime }).
\end{equation*}
\end{itemize}
\item intuition: entrepreneurs take on risky projects with higher rewards.
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Period 1 overview}
\begin{itemize}
\item Entrepreneurs endowed with $K$ units of capital and $B$ units of debt
(owed to workers)

\item Entrepreneurs produce according to
\begin{equation*}
Y=AK^{\alpha }\ell ^{1-\alpha }.
\end{equation*}

\item Capital accumulation and resource constraints%
\begin{align}
k^{\prime }& =K+\omega i,  \tag{CA} \\
i& =Y-c_{w}-c_{e}.  \notag
\end{align}%
$\omega $ is an idiosyncratic, entrepreneur-specific, shock with aggregate
value of unity (b/c risk neutral)

\begin{itemize}
\item distributed $\Phi \left( \omega \right) $
\item observed after the entrepreneur chooses $i$ (investment)
\item distinction between workers and entrepreneurs
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Period 2 overview}
\begin{itemize}
\item Capital (produced by entrepreneurs in first period) is sold to
entrepreneurs and worker-owned firms%
\begin{equation*}
k^{\prime }=k_{w}^{\prime }+k_{e}^{\prime }
\end{equation*}

\item Entrepreneurs produce according to
\begin{equation*}
y_{e}^{\prime }=a^{\prime }k_{e}^{\prime },
\end{equation*}

\item Worker-owned firms produce according to
\begin{equation*}
y_{w}^{\prime }=a^{\prime }G(k_{w}^{\prime }),
\end{equation*}%
where $G(\cdot )$ is strictly concave, with $G^{\prime }\left( 0\right) =1$

\item Entrepeneur capital more productive, but risky ($\omega$ unknown).

\item Assume that $\beta a^{\prime }>1,$ implying that can raise utility by
postponing consumption through investment
\end{itemize}
\end{frame}



\begin{frame}\frametitle{Worker's problem}
\begin{align*}
\underset{c_{w},\ell ,k_{w}^{\prime },b^{\prime }}{\max }& \left\{ c_{w}-%
\frac{\ell ^{2}}{2}+\beta c_{w}^{\prime }\right\} \\
s.t.\quad & B+w\ell =c_{w}+\frac{b^{\prime }}{R}+qk_{w}^{\prime }, \\
& c_{w}^{\prime }=a^{\prime }G(k_{w}^{\prime })+b^{\prime }, \\
& c_{w}\geq 0;\quad c_{w}^{\prime }\geq 0,
\end{align*}%
where

\begin{itemize}
\item $w$ is the real wage

\item $R$ is the gross interest rate earned on bonds

\item $q$ is the price of capital in terms of consumption/output
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Solving worker's problem}
\begin{itemize}
\item First order conditions for workers' problem (no uncertainty)%
\begin{eqnarray*}
\ell &=&w(1+\lambda ), \\
(1+\lambda )q &=&\beta a^{\prime }G^{\prime }(k_{w}^{\prime }), \\
1+\lambda &=&\beta R,
\end{eqnarray*}%
where $\lambda $ is the multiplier on $c_{w}\geq 0$ (Note: $\beta a^{\prime
}>1\Rightarrow c_{w}^{\prime }>0$ since agents will always want to transfer
consumption forward.)
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Entrepreneurs' problem}
\begin{align}
\underset{c_{e},\ell ,i,k_{e}^{\prime },b^{\prime }}{\max }& E\left\{
c_{e}+\beta c_{e}^{\prime }\right\}  \notag \\
s.t.\quad & qk_{e}^{\prime }=V+qK+(q\omega -1)i+\frac{b^{\prime }}{R}%
-B-c_{e},  \tag{EFBC} \\
& V=AK^{\alpha }\ell ^{1-\alpha }-w\ell ,  \notag \\
& c_{e}^{\prime }=a^{\prime }k_{e}^{\prime }-b^{\prime },  \notag \\
& c_{e}\geq 0;\quad c_{e}^{\prime }\geq 0;\quad i\geq 0.  \notag
\end{align}

\begin{itemize}
\item $c_{e}$, $b^{\prime }$ and $k_{e}^{\prime }$ chosen after $\omega $ is
realized

\item $i$ chosen before $\omega $ is realized
\item $B$: initial bonds.
\item $V$: firm profits.
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Solving entrepreneur's problem}
\begin{itemize}
\item First order conditions for entrepreneurs%
\begin{eqnarray*}
w &=&(1-\alpha )AK^{\alpha }\ell ^{-\alpha } \\
qE(\omega ) &\leq &1,\quad (=\text{if }i>0), \\
(1+\gamma )q &=&\beta a^{\prime }, \\
1+\gamma &=&\beta R,
\end{eqnarray*}%
where $\gamma $ is the multiplier on $c_{e}\geq 0$
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Market clearing}
\begin{itemize}
\item F.O.C. on $b^{\prime }$ for entrepreneur and worker imply $\gamma
=\lambda $

\begin{itemize}
\item Non-negativity constraint on period 1 consumption is either binding or
not for both entrepreneur and worker
\end{itemize}

\item Combine the F.O.C.'s for labor $\left( \ell \right) $ for worker
(labor supply) and entrepreneur (labor demand) to get
\begin{eqnarray*}
w &=&(1-\alpha )AK^{\alpha }[w(1+\lambda )]^{-\alpha } \\
&=&(1-\alpha )^{\frac{1}{1+\alpha }}A^{\frac{1}{1+\alpha }}K^{\frac{\alpha }{%
1+\alpha }}(1+\lambda )^{\frac{-\alpha }{1+\alpha }} \\
\ell &=&w(1+\lambda ) \\
&=&(1-\alpha )^{\frac{1}{1+\alpha }}A^{\frac{1}{1+\alpha }}K^{\frac{\alpha }{%
1+\alpha }}(1+\lambda )^{\frac{1}{1+\alpha }},
\end{eqnarray*}%
so that labor, current output, and profits $\left( V=AK^{\alpha }\ell
^{1-\alpha }-w\ell =\alpha Y\right) $ \ are all increasing in current
productivity, $A$, and the multiplier $\lambda $
\end{itemize}
\end{frame}

% \subsubsection{Frictionless market, no uncertainty $\left( \protect\omega %
% =1\right) $}

\begin{frame}\frametitle{Frictionless market, no uncertainty ($\omega = 1$)}
\begin{equation*}
\beta a^{\prime }=\beta a^{\prime }G^{\prime }(k_{w}^{\prime }),
\end{equation*}%
\begin{itemize}
\item so that $k_{w}^{\prime }=0$ and $k^{\prime }=k_{e}^{\prime }$
\item $\rightarrow $ entrepreneurs use all the capital
\item why? entrepreneurs use risky production tech. with higher returns
\item here no risk.
\end{itemize}
\end{frame}

% \subsubsection{Uncertainty}

% \paragraph{Good outcome $E\protect\omega =1$}

\begin{frame}\frametitle{Uncertainty, high returns $E\protect\omega =1$}
\begin{itemize}
\item Investment occurs $\left( i>0\right) ,$ and
\begin{eqnarray*}
q &=&\frac{1}{E(\omega )}=1 \\
\lambda &=&\beta a^{\prime }q^{-1}-1 \\
&=&\beta a^{\prime }-1\geq 0,
\end{eqnarray*}%
so that labor, current output and profits are all increasing in future
productivity, $a^{\prime }$

\item Additionally%
\begin{equation*}
c_{w}=c_{e}=0
\end{equation*}%
\begin{equation*}
i=Y
\end{equation*}%
so that all output is optimally invested
\end{itemize}
\end{frame}

% \paragraph{Bad outcome $E\protect\omega =0$}

\begin{frame}\frametitle{Uncertainty, low returns $E\protect\omega =0$}
\begin{itemize}
\item No investment occurs

\item Consumption for both worker and entrepreneur in first period must be
positive satisfying resource constraint implying that
\begin{equation*}
\lambda =\gamma =0
\end{equation*}

\item Actual price of capital
\begin{equation*}
q=\beta a^{\prime }
\end{equation*}

\item No benefits of transferring resources forward due to low expected
price for capital

\item Inefficient outcome because output not transferred forward due to
expectations, {\color{red} implying that expectations have real effects}
\end{itemize}
\end{frame}


\section{Microfounding financial frictions}

\begin{frame}\frametitle{Costly state verification}
\begin{itemize}
\item Asymmetric information between borrowers and lenders

\item Sources of funds:\ $i=N+D$, where
\begin{equation*}
N=qK+V-B
\end{equation*}%
is the entrepreneur's net worth

\item $D\geq 0$ is a within-period loan (no interest)

\item $D$ is repaid immediately after $\omega $ is realized and capital is
produced
\item $qK$: value of capital
\item $V$: profits
\item $B$: bonds (debt).
\end{itemize}
\end{frame}


\begin{frame}\frametitle{Incomplete information}
\begin{itemize}
\item Entrepreneur observes realization of $\omega $

\item Lender observes $\omega $ only by paying cost $\mu i$

\item Solution (Townsend, 1979): standard debt contract

\begin{itemize}
\item If the borrower pays $(1+r_{k})D$, lender doesn't check $\omega $

\item If the borrower pays less than $(1+r_{k})D$, lender pays cost and
verifies $\omega $
\end{itemize}

\item With incomplete information, equity requires verification every period
and is therefore less efficient
\end{itemize}
\end{frame}

% \subsubsection{Contract terms}

\begin{frame}\frametitle{Contract}
\begin{itemize}
\item Verification ensures that borrower defaults only if $\omega <\tilde{%
\omega},$ where%
\begin{align*}
\widetilde{\omega }qi& =\widetilde{\omega }q(N+D)=(1+r_{k})D, \\
\Rightarrow \widetilde{\omega }& =\widetilde{\omega }(r_{k},q,N,D)=\frac{%
1+r_{k}}{q}\left( \frac{D}{N+D}\right) .
\end{align*}
\item i.e., only default if really bad shock.
\item Competitive lenders implies a zero-profit condition%
\begin{equation*}
\hspace{-0.75cm}\int_{0}^{\widetilde{\omega }(r_{k},q,N,D)}\hspace{0.2in}%
\hspace{-1.25cm}\;(\omega -\mu )q(N+D)\;d\Phi (\omega )+\int_{\widetilde{%
\omega }(r_{k},q,N,D)}^{\infty }\hspace{-1.25cm}\;\hspace{0.2in}%
(1+r_{k})D\;d\Phi (\omega )=D.
\end{equation*}%
defines $r_{k}(q,N,D)$ and $\widetilde{\omega }(q,N,D)=$ $\widetilde{\omega }%
(r_{k}(q,N,D),q,N,D)$
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Contract II}
\begin{itemize}
\item Competitive lenders implies a zero-profit condition%
\begin{equation*}
\hspace{-0.75cm}\int_{0}^{\widetilde{\omega }(r_{k},q,N,D)}\hspace{0.2in}%
\hspace{-1.25cm}\;(\omega -\mu )q(N+D)\;d\Phi (\omega )+\int_{\widetilde{%
\omega }(r_{k},q,N,D)}^{\infty }\hspace{-1.25cm}\;\hspace{0.2in}%
(1+r_{k})D\;d\Phi (\omega )=D.
\end{equation*}%
defines $r_{k}(q,N,D)$ and $\widetilde{\omega }(q,N,D)=$ $\widetilde{\omega }%
(r_{k}(q,N,D),q,N,D)$

\item Solve for interest rate on debt%
\begin{equation*}
(1+r_{k})=\frac{1-\int_{0}^{\widetilde{\omega }}(\omega -\mu )q(\frac{N+D}{D}%
)\;d\Phi (\omega )}{\int_{\widetilde{\omega }}^{\infty }d\Phi (\omega )}
\end{equation*}

\item Smaller $N$ $\rightarrow$ smaller repayments in default
\item $\rightarrow$ and a larger $\widetilde{\omega }$,
\item $\rightarrow$ probability of default $\uparrow$.
\item $\rightarrow$ Interest rate on debt increases.
\item Real effects on economy.
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Optimal loans}
\begin{itemize}
\item Lenders compete to make loans, offer contracts that maximize
entrepreneur's profits

\item The optimal loan $D\left( N,q\right) $ is picked to solve%
\begin{align*}
\hspace{-0.5cm}& \underset{D}{\max }\int_{\widetilde{\omega }%
(q,N,D)}^{\infty }\Pi (\omega ,q,N,D)\;d\Phi (\omega ), \\
& \Pi (\omega ,q,N,D)=\omega q(N+D)-[1+r_{k}(q,N,D)]D.
\end{align*}

\item This yields%
\begin{equation*}
D\left( N,q\right)
\end{equation*}%
\begin{equation*}
r_{k}(q,N)=r_{k}(q,N,D(q,N))
\end{equation*}%
\begin{equation*}
\widetilde{\omega }(q,N)=\widetilde{\omega }(q,N,D(q,N))
\end{equation*}%
\begin{equation*}
\Pi (\omega ,q,N)=\max \Big\{0,\Pi (\omega ,q,N,D(q,N))\Big\}
\end{equation*}%
(don't enter or do)
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Savings}
\begin{itemize}
\item Now, entrepreneur can save after realizing $\omega$.
\item The entrepreneur's hiring decision is same as before, and is independent of
his other decisions (given $\omega $)

\begin{itemize}
\item After $\omega $ has been realized, the entrepreneur solves%
\begin{align}
\underset{c_{e},k_{e}^{\prime },b^{\prime }}{\max }& \left\{ c_{e}+\beta
c_{e}^{\prime }\right\}  \notag \\
s.t.\quad & qk_{e}^{\prime }=\Pi (\omega ,q,N)+\frac{b^{\prime }}{R}-c_{e},
\tag{EFBC2} \\
& c_{e}^{\prime }=a^{\prime }k_{e}^{\prime }-b^{\prime },  \notag \\
& c_{e}\geq 0;\quad c_{e}^{\prime }\geq 0.  \notag
\end{align}
\end{itemize}

\item The worker's problem is unchanged
\end{itemize}
\end{frame}

\begin{frame}
\begin{itemize}
\item First-order conditions
\begin{eqnarray*}
(1+\lambda )q &=&\beta a^{\prime }G^{\prime }(k_{w}^{\prime }), \\
1+\lambda &=&\beta R, \\
(1+\gamma )q &=&\beta a^{\prime }, \\
1+\gamma &=&\beta R,
\end{eqnarray*}%
where $\lambda $ and $\gamma $ are multipliers
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Savings scenarios}
\begin{itemize}
\item Scenario 1: $N$ is large enough to support $i=Y.$ Similar to
frictionless case

\item Scenario \ 2: $N$ is smaller so that $i<Y$

\item Focus on scenario 2
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Scenario 2}
\begin{itemize}
\item With $i<Y,$ there is period-1 consumption $\Rightarrow \lambda =\gamma
=0$ and%
\begin{eqnarray*}
q &=&\beta a^{\prime }>1, \\
R &=&\beta ^{-1},
\end{eqnarray*}

\item Effect of increasing $A$ (aggregate shock period-1)

\begin{itemize}
\item $N$ increases $\Rightarrow i=N+D$ increases, implying that $y^{\prime
} $ increases

\item Unless $\Delta i>\Delta Y,$ increase in $y^{\prime }$ is less than in
the frictionless environment
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Dynamics}
\begin{itemize}
\item Effects of increasing $a^{\prime }$ (aggregate shock period-2)

\begin{itemize}
\item $q=\beta a^{\prime }$ increases

\item $q$ increases $\rightarrow $ investment increases $\rightarrow
$ $y^{\prime }$ increases

\item $q$ increases $\rightarrow $ investment more profitable $%
\rightarrow $ $D$ increases $\rightarrow $ $y^{\prime }$ increases
\end{itemize}

\item Quantitative performance

\begin{itemize}
\item Frictions dampen initial responses to TFP shocks

\item Carlstrom and Fuerst (AER 1997): in a multi-period model, net worth
accumulation generates a hump-shaped impulse response function

\item Capital adjustment costs that affect $q$ increase propagation
(Bernanke, Gertler and Gilchrist, 1999)
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Conclusion}
\begin{itemize}
\item Today: Financial frictions
\item Next time: maybe more financial frictions or income fluctuation problem/heterogeneous agents.
\end{itemize}
\end{frame}

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


\end{document}
