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% ----------------------------------------------------------------------------------------
%	TITLE PAGE
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\title[]{Quantitative Macro-Labor:\\ Business Cycles and Search} % 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{Fall 2024} % 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{Announcements}
  \begin{itemize}
  \item Today: BRE with more sources of heterogeneity.
  \item Herkenhoff (2019): Search with business cycles and realistic credit markets.
  \item Data project due 11/5 (three-ish weeks), presentations same week (probably).
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Incorporating Heterogeneity into Search Models}
  \begin{itemize}
  \item What makes the Block Recursive Model tractable?
    \begin{enumerate}
    \item Simple general equilibrium.
    \item Segment markets appropriately and the rest is easy.
    \end{enumerate}
  \item What makes other GE search models tricky?
    \begin{enumerate}
    \item Vacancy creation decision.
    \item On-the-job search decision.
    \end{enumerate}
  \item Today: one more extension of BRE.
  \end{itemize}
\end{frame}

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

\section{Search for Credit} % 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{Herkenhoff (2019)}
  \begin{itemize}
  \item What is the impact of credit constraints on business cycles?
  \item Basic model:
    \begin{enumerate}
    \item Menzio \& Shi (2011), life-cycle version.
    \item + endogenous credit constraints.
    \item + search for lenders.
    \end{enumerate}
  \item Credit can constrict in response to labor market shocks and aggregate conditions.
  \end{itemize}
\end{frame}


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

\begin{frame}
  \frametitle{Regularities (2015 WP version)}
\includegraphics[width=\textwidth]{./Herkenhoff2015-1.png}
  \begin{itemize}
  \item Credit expanded until the Great Recession.
  \end{itemize}
\end{frame}


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

\begin{frame}
  \frametitle{Model Environment}\label{environment}
  \begin{itemize}
  \item Life-cycle model: age discrete, indexed by $t$; exit at $T+1$.
  \item Agents:
    \begin{itemize}
    \item Employed and unemployed workers.
    \item Matched and unmatched firms.
    \item Credit lending agencies.
  \end{itemize}
  \item Technology:
    \begin{itemize}
    \item Frictional matching in labor \& credit markets.
    \item Endogenous credit constraints.
    \end{itemize}
  \item Initial heterogeneity:
    \begin{itemize}
    \item Initial wealth ($a_{0}$) and employment status.
    \end{itemize}
  % \item Timing: \hyperlink{timing}{\beamergotobutton{here}}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Agents}\label{agents}
  \begin{itemize}
  \item Risk-averse workers indexed by $(e, a, w, b, z, t)$
    \begin{itemize}
    \item Employed ($w$), unemployed w/ UI ($z$).
    \item Search for credit among offers. Make default decision.
    \item With credit: borrow up to $b'$. Without, $b'\geq 0$.
    \end{itemize}
  \item Continuum of profit maximizing employers:
    \begin{itemize}
    \item Risk neutral.
    \item Post vacancies that specify wage ($w$).
    \end{itemize}
  \item Continuum of lenders:
    \begin{itemize}
    \item Risk neutral. Pay $r_{F}$ on deposits.
    \item Post offers for credit to potential borrowers with schedule $q(w,b')$.
    \end{itemize}
  \item Type-distribution $\phi' = \Phi(\phi)$ (suppressed throughout).
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Search and Matching Technology}\label{searchandmatching}
  \begin{itemize}
  \item Directed search (Moen, 1997):
    \begin{itemize}
    \item Labor submarket: homogeneous workers $(e, a, b, z, t)$ and firms $(w)$
    \item Credit submarket: homogeneous workers $(e, a, b, z, t)$ and firms $(q)$
    \end{itemize}
  \item Matching technology:
    \begin{itemize}
    \item \# of matches in submkt: $M_{t} = M(u_{t},v_{t})$ (CRS).
    % \item Definitions:
    %   \begin{itemize}
      \item Submarket tightness: $\theta_{t}(\cdot) = \frac{v_{t}}{u_{t}}$
      \item Worker finding rate: $q(\theta_{t}) = \frac{M(u_{t},v_{t})}{v_{t}}$
      \item Job finding rates: $p(\theta_{t}) = \frac{M(u_{t},v_{t})}{u_{t}} = \theta_{t}q(\theta_{t})$
      % \end{itemize}
    \end{itemize}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Credit Searcher's Problem}\label{unemployedworkerssearching}
  \begin{itemize}
  \item States: $s_{U} = (z, b)$, $s_{C}'=S_{N}' = (z, b)$, $z = \gamma w$
  \item Each period, borrowers search for lender:
    \begin{itemize}
    \item Meet w/ prob. $A\psi(\theta_{t}(s_{C}'))$.
    \item Continue w/ value $U_{t}^{C}(s_{C}')$ if offered job.
    \item Continue w/ value $U_{t}^{N}(s_{N}')$ if no offer.
    \end{itemize}
  \item Bellman for credit searcher:
    \begin{equation*}
      U_{t}(s_{U}) =A\psi(\theta_{t}(s_{C}'))U_{t}^{C}(s_{C}') + (1 - A\psi(\theta_{t}(s_{C}')))U_{t}^{N}(s_{N}')
    \end{equation*}
  \item The lending market tightness will be defined after the problem.
  \end{itemize}
\end{frame}

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



\begin{frame}
  \frametitle{Worker's Problem with Lender}\label{unemployedworkersconsumption}
  \begin{itemize}
  \item States:
    \begin{itemize}
    \item Unemp. w/ Credit: $s_{U} = (z, b)$, $s_{U}' = (z, b')$, $s_{E}' = (w, b')$
    \end{itemize}
  \item Policies:
      \begin{itemize}
      \item Consume \& save s.t. $b'\geq\underline{b}$.
      \item Default fraction $D\in[0,1]$
      \end{itemize}
  \item Value of unemployment (w/ credit):
\begin{align*}
      U_{t}^{C}(s_{U}) &= \max_{b\in\mathbbm{B},D\in[0,1]}u(c) - x(D) + \eta\\&+ \beta (1 - S(D))R_{t + 1}^{C} + S(D)R_{t + 1}^{N}\\
      &\text{s.t. } c + q_{U,t}(z,b')b'\leq z + (1 - D)b \\
      &y'\sim F(y'|y)
    \end{align*}
\begin{align*}
R_{t + 1}^{X} &= E[\max_{w\in\mathbbm{W}}p(\theta_{t + 1}(s_{E}'))W_{t + 1}^{X}(s_{E}') \\&+ (1 - p(\theta_{t + 1}(s_{E}')))U_{t + 1}^{X}(s_{U}'))]
\end{align*}
  \item No OTJS, but employed similar.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Worker's Problem without Lender}\label{unemployedworkersconsumption}
  \begin{itemize}
  \item States:
    \begin{itemize}
    \item Unemp. w/o Credit: $s_{U} = (z, b)$, $s_{U}' = (z, b')$, $s_{E}' = (w, b')$
    \end{itemize}
  \item Policies:
      \begin{itemize}
      \item Consume \& save s.t. $b'\geq 0$.
      \item Default fraction $D\in[0,1]$
      \end{itemize}
  \item Value of unemployment (w/o credit):
\begin{flalign*}
      U_{t}^{N}(s_{U}) &= \max_{b\in\mathbbm{B},D\in[0,1]}u(c) \\&+ \beta E[\max_{w\in\mathbbm{W}}p(\theta_{t + 1}(s_{E}'))W_{t + 1}(s_{E}') \\&+ (1 - p(\theta_{t + 1}(s_{E}')))U_{t + 1}(s_{U}'))]\\
      &\text{s.t. } c + \frac{1}{1 + r_{F}}b' \leq z + (1 - D)b \\
      &y'\sim F(y'|y)
    \end{flalign*}
  \item No OTJS, but employed similar.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Savings and Lendings Institutions}\label{firms}
  \begin{itemize}
  \item Two types:
    \begin{enumerate}
    \item Savings institutions: offer risk free rate $r_{F}$ to everyone $b > 0$.
    \item Credit institutions: offer schedule or prices $q$ based on credit requested and worker state.
    \end{enumerate}
  \item States: $s_{q} = (e, w, b)$
  \item Matched firms:
    \begin{itemize}
    \item Price $q$, quantity $\hat{b}$;
    \item Worker defaults on fraction $D$;
    \item Must borrow at world risk free rate.
    \end{itemize}
  \item Value of one-period lending relation:
    \begin{align}
      Q_{t}(s_{q}) &= q(s_{q})\hat{b} - \frac{1}{1 + r_{F}}E[(1 - D)\hat{b}]
    \end{align}
    \item Worker decisions: $D, b'$.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Credit Market Free Entry}\label{freeentry}
  \begin{itemize}
  \item States: $s_{q} = (e, w, b)$
  \item Normally, free entry used in labor market.
  \item Here, lenders compete away profits.
    \item Two types:
      \begin{itemize}
      \item Paying risk-free rate.
      \item Lending at price $q$ at servicing fee $\tau$.
      \end{itemize}
    \item Free Entry ($V_{t}(s_{q}) = 0$):
      \begin{align*}
        q(\theta_{t}(s_{q})) = \begin{cases} \frac{E[(1 - D(s_{q}))]}{1 + r_{F} + \tau} & \hat{b}\in\mathbbm{B}_{-}\\
0 & \hat{b}\in\mathbbm{B}_{-}, D>0\\
          \frac{1}{1 + r_{F}} & \hat{b}\in\mathbbm{B}_{+}
          \end{cases}
      \end{align*}
    \item Contracts are offered to each type $s_{q}$.
  \end{itemize}
\end{frame}


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

\begin{frame}
  \frametitle{Firms}\label{firms}
  \begin{itemize}
  \item States: $s_{J} = (w), s_{J}' = (w)$
  \item Why not $b'$?
  \item Matched firms:
    \begin{itemize}
    \item Net profits $y_{t} - w_{t}$;
    \item separate exog. w/ prob. $\delta$;
    \item continue w/ value $J_{t + 1}(s_{J}')$
    \end{itemize}
  \item Value of filled vacancy with age-$t$ type-$s_{J}$ worker:
    \begin{flalign*}
      \hspace{-5mm}J_{t}(s_{J}) &=  y - w + \beta E[(1 - \delta)J_{t + 1}(s_{J}')]\\
      y' &= F(y'|y)
    \end{flalign*}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Free Entry and Equilibrium Job-Finding Rates}\label{freeentry}
  \begin{itemize}
  \item States: $s_{J} = (w)$
    \item Unmatched firms:
      \begin{itemize}
      \item Pay $\kappa_{L}$ to post (profitable) vacancies.
      \item Match w/ prob. $f(\theta_{t}(s_{J}))$.
      \end{itemize}
    \item Value of vacancy with age-$t$ worker paying wage $w$:
      \begin{equation*}
        V_{t}(s_{J}) = -\kappa_{L} + f(\theta_{t}(s_{J}))J_{t}(s_{J})
      \end{equation*}
    \item Free Entry ($V_{t}(s_{J}) = 0$):
      \begin{align*}
        f(\theta_{t}(s_{J})) &= \frac{\kappa_{L}}{J_{t}(s_{J})}\\
        \theta_{t}(s_{J}) &= q^{-1}(\frac{\kappa_{L}}{J_{t}(s_{J})})
      \end{align*}
  \end{itemize}
\end{frame}


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

\begin{frame}
  \frametitle{Equilibrium}\label{equilibrium}
A {\it Block Recursive Equilibrium} (BRE) in this model is a set of value functions, $U_{t}, W_{t}, U_{t}^{C}, U_{t}^{N}, W_{t}^{C}, W_{t}^{N}, J_{t}^{L}, V_{t}^{L}, J_{t}^{C}, V_{t}^{C}$, associated policy and market tightness functions, $b', c, w'$, $\theta_{t}^{L}$, and $\theta_{t}^{C}$, which satisfy
\begin{enumerate}
\item The policy functions solve the workers problems.
\item $\theta_{t}(w)^{L}$ satisfies the free entry condition for all submarkets $(w)$ in the labor market.
\item $\theta_{t}(e,w,b)^{C}$ satisfies the free entry condition for all submarkets $(w)$ in the credit market.
\item The aggregate law of motion is consistent with all policy functions.
\end{enumerate}\vspace{5mm}
\end{frame}

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

\section{Calibration} % 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{Calibration (Stochastic Steady State)}
  \centering\includegraphics[width=\textwidth]{./Herkenhoff2019-Calibration.png}
\end{frame}

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

\begin{frame}
  \frametitle{Fit}
  \includegraphics[width=\textwidth]{./Herkenhoff2019-SMM.png}
\end{frame}

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

\begin{frame}
  \frametitle{Fit}
  \includegraphics[width=\textwidth]{./Herkenhoff2019-NonTargeted.png}
\end{frame}

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

\section{Findings} % 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{How Does Credit Affect Employment?}
  \begin{itemize}
  \item Four recent business cycles: 1980 (exp), 1990 (rec), 2001 (rec), and 2007 (rec).
  \item Input path of aggregate shocks \& credit access.
  \item Quantify impact of credit on aggregates.
  \end{itemize}
\end{frame}

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

% \begin{frame}
%   \frametitle{The 1990 Recession (five points if you knew there was a 1990 recession)}
%   \includegraphics[width=\textwidth]{./Herkenhoff2015-5.png}
% \end{frame}

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

\begin{frame}
  \frametitle{2001 Transition Path}
  \includegraphics[width=\textwidth]{./Herkenhoff2019-2.png}
\end{frame}

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

\begin{frame}
  \frametitle{Job-Finding and Employment 2001 (2015 WP version)}
  \includegraphics[width=\textwidth]{./Herkenhoff2015-3.png}
\end{frame}

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

\begin{frame}
  \frametitle{Consumption Risk during 2001 (2015 WP version)}
  \includegraphics[width=\textwidth]{./Herkenhoff2015-4.png}
\end{frame}

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

% \begin{frame}
%   \frametitle{The Great Recession}
%   \includegraphics[width=\textwidth]{./WinterhasComeGreatRecession.jpg}
% \end{frame}

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

\begin{frame}
  \frametitle{Inputs and Outputs}
  \includegraphics[width=\textwidth]{./Herkenhoff2019-6.png}
\end{frame}

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

\begin{frame}
  \frametitle{Transition Experiments (2015 WP version)}
  \includegraphics[width=\textwidth]{./Herkenhoff2015-7.png}
\end{frame}

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

\begin{frame}
  \frametitle{Employment Impulse Responses (2015 WP version)}
  \includegraphics[width=\textwidth]{./Herkenhoff2015-8.png}
\end{frame}

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

\begin{frame}
  \frametitle{Consumption Risk Impulse Responses (2015 WP version)}
  \includegraphics[width=\textwidth]{./Herkenhoff2015-9.png}
\end{frame}

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

\begin{frame}
  \frametitle{Numerical Findings}
  \includegraphics[width=\textwidth]{./Herkenhoff2015-Findings.png}
\end{frame}

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

\section{Conclusion}
% ------------------------------------------------


\begin{frame}
  \frametitle{Conclusion}
  \begin{itemize}
  \item Next time: Solution techniques.
  \item Data projects due 11/5 with presentations to follow!
  \end{itemize}
\end{frame}


\end{document}
