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%----------------------------------------------------------------------------------------
%	TITLE PAGE
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\title[]{Macro II Extra topics:\\ Frictions and Heterogeneity with the Simplicity of the Representative Agent: The Block Recursive Equilibrium} % 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 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 One more class after today.
  \item Today:
    \begin{enumerate}
    \item Discuss Menzio and Shi (2011)
    \item Block Recursive Equilibrium
    \item Why this is useful.
    \end{enumerate}
  \item HW due this week and next.
  \item Last class: probably not review, open to suggestions.
  \end{itemize}
\end{frame}

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


\begin{frame}
\frametitle{Motivation}
\begin{itemize}
\item Search and matching models of the labor market can be hard to solve:
\begin{enumerate}
\item Vacancy posting firm: needs to know the distribution of workers \& their reservation/application strategies.
\item Searching worker: needs to know the number of vacancies to set reservation/application strategies.
\item Potentially very complicated fixed point problem.
\end{enumerate}
\item Problem becomes much harder with heterogeneity:
  \begin{enumerate}
  \item Vacancy posting firm needs to know the type distribution of workers, as well as the reservation/application strategy by types.
  \item Workers need to know the number of vacancies, i.e., the type distribution of workers.
  \end{enumerate}
\item Menzio and Shi: specify prices in such a way that they do not depend on the distribution.
\end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{Menzio and Shi (2011)}
\begin{itemize}
\item Empirical goal: match business cycle ``regularities'' about worker flows:
  \begin{enumerate}
  \item UE (unemployment-employment) rate: 42\% monthly
  \item EU rate: 2.6\% monthly
  \item EE rate (OTJ transitions): 2.9\% monthly
  \item Substantial fluctuations over business cycle
  \end{enumerate}
\end{itemize}
\includegraphics[width=\textwidth]{./Tab1.png}
\end{frame}

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

\begin{frame}
\frametitle{Menzio and Shi (2011)}
\begin{itemize}
\item To match worker flow regularities, need:
  \begin{enumerate}
  \item On-the-job search
  \item Productivity fluctuations
  \item Match heterogeneity with endogenous separation
  \end{enumerate}
\item Problem:
  \begin{enumerate}
  \item Worker on-the-job reservation/application strategy impacted by current and future productivity \& worker distributions.
  \item $\rightarrow$ vacancy posting impacted by expected future productivity, and worker distributions.
  \item $\rightarrow$ equilibrium hard to solve out of steady-state.
  \end{enumerate}
\item Here:
  \begin{enumerate}
  \item Specify equilibrium so that vacancy posting and search behavior do not depend on distribution of workers.
  \item Becomes a decision theory problem.
  \item ``...is just as easy as solving the planner's problem in a representative agent model''
  \end{enumerate}
\end{itemize}
\end{frame}

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

% \section{Background} % 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{This is the first page of the second section}
% \begin{itemize}
% \item Here's an equation:
% \begin{equation}
% p_{it}^{Y} = \alpha + \beta p_{it}^{X} + \epsilon_{it}
% \end{equation}
% \item It equals one billllionnnn dollars
% \begin{enumerate}
% \item sharks with laser beams
% \end{enumerate}
% \end{itemize}
% \end{frame}

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

\section{Model} % 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{Environment}
\begin{itemize}
\item Workers (risk-neutral):
  \begin{enumerate}
  \item Infinitely-lived, can be employed or unemployed.
  \item Directed search on and off the job (otjs efficiency $\lambda_{e}<= 1$).
  \end{enumerate}
\item Firms (risk-neutral):
  \begin{enumerate}
  \item Productivity of job (match): $y + z$.
  \item $y$ is an evolving aggregate component $y'\sim\Phi(y'|y)$.
  \item $z$ is a fixed match-specific component (iid between matches).
  \end{enumerate}
\item Jobs (matched worker-firm pair):
  \begin{enumerate}
  \item Workers apply for vacancies posted by unmatched firms.
  \item Signal of match quality: $s = z$ w/ prob. $\alpha$, $s\sim f(z)$ w/ $1 - \alpha$.
  \item Separation rate $d$ = exog. $\delta$ and endog. (OTJS + fired)
  \end{enumerate}
\item Discrete time; common discount factor $\beta$.
\end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{Directed Search \& Posting}
\begin{itemize}
\item Canonical random search model (Pissarides, 1985):
  \begin{enumerate}
  \item Workers ``randomly'' meet firms.
  \item Terms of employment are not settled until after matched.
  \item Some meetings not accepted.
  \end{enumerate}
\item Directed search (Moen, 1997; Shimer 1996):
  \begin{enumerate}
  \item Terms of employment announced prior to match.
  \item Workers ``direct'' their search to preferred terms.
  \item No ``inefficient unemployment'': all meetings accepted.
  \end{enumerate}
\item Key terminology:
  \begin{enumerate}
  \item Submarket ``tightness'': $\theta = \frac{v}{u}$.
  \item Submarket indexed by worker/firm state and terms.
  \item Contact rate of workers: $p(\theta)$.
  \item Contact rate of firms: $q(\theta) = \frac{p(\theta)}{\theta}$.
  \end{enumerate}
\end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{Contracts}
\begin{itemize}
\item Firms offer promised value $x$, and reservation signal, $r$.
\item Contracts are complete:
  \begin{enumerate}
  \item Specify separation threshold $d(z,y)$ for each $(z, y)$
  \item Specify submarket for OTJS: $(x, r)$
  \item Maximize joint value of match
  \end{enumerate}
\item Equivalent to firm setting wage as function of tenure and productivity
\item \& worker picking separation threshold.
\end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{Timing}
\begin{enumerate}
\item Aggregate productivity, $y$ realized.
\item Jobs are destroyed with probability $d \in [\delta, 1]$.
\item Workers search, firms offer contracts.
\item Workers and firms match, draw productivity, $z$, see signal, $s$.
\item Consume and produce.
\end{enumerate}
\end{frame}

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

\begin{frame}
\frametitle{Unemployed Decentralized Problem}
\begin{itemize}
\item Submarkets are indexed by promised utility, $x$, and the signal required to maintain employment, $r$, $s >= r$.
\item $\psi$ is aggregate productivity \& worker distributions.
\item Bellman Equation for search sub-period:
  \begin{align}
    D(x, r, V, \psi) = p(\theta(x, r, \psi))m(r)(x - V)
  \end{align}
\item Unemployed Bellman Equation for consumption sub-period:
  \begin{align}
    V_{u}(\psi) = b + \beta E[\max_{x,r}\{V_{u}(\hat{\psi}) + \lambda_{u}D(x, r, V(\hat{\psi}), \hat{\psi})\}]
  \end{align}
\end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{Matched Decentralized Problem}
\begin{itemize}
\item Transferability of utility $\rightarrow$ surplus sum of worker \& firm value.
\item Matched Bellman Equation for consumption sub-period:
  \begin{align}
    V_{e}(z,\psi) &= y + z + \beta E\big[\max_{d,x,r}\{d V_{u}(\hat{\psi}) \nonumber\\&+ (1 - d)[V_{e}(z,\hat{\psi}) + \lambda_{e}D(x, r, V(z,\hat{\psi}), \hat{\psi})]\}\big]
  \end{align}
\item $d(z,y) = 1$ iff $z < r_{d}(y)$: unemp val. $>$ cont. val.
\item Bellman Equation for search sub-period:
  \begin{align}
    D(x, r, V, \psi) = p(\theta(x, r, \psi))m(r)(x - V)
  \end{align}
\item Unemployed Bellman Equation for consumption sub-period:
  \begin{align}
    V_{u}(\psi) = b + \beta E[\max_{x,r}\{V_{u}(\hat{\psi}) + \lambda_{u}D(x, r, V(\hat{\psi}), \hat{\psi})\}]
  \end{align}
\end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{Vacancy Creation Condition}
\begin{itemize}
\item Unmatched firms must open vacancies at cost $\kappa$ to find workers.
\item Expected profits from opening a vacancy (vacancy creation):
  \begin{align}
    \hspace{-10mm}\small V_{v}(x,r,\psi) &= \underbrace{-\kappa}_{Cost} \nonumber\\&+ q(\theta(x,r,\psi))\sum_{s\geq r}\{\underbrace{\alpha V_{e}(s,\psi)}_{Correct\;Signal} + \underbrace{(1 - \alpha)E_{z}[V_{e}(z,\psi) - x]\}f(s)}_{Random\;Signal}
\end{align}
\item If $\alpha = 1$: pure ``inspection'' good
\item If $\alpha = 0$: pure ``experience'' good
\item No learning: $z$ known immediately after employment.
\end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{Free Entry Condition}
\begin{itemize}
% \item Expected profits from opening a vacancy (vacancy creation):
%   \begin{align}
%     \hspace{-10mm}\small V_{v}(x,r,\psi) &= -\kappa \nonumber\\&+ q(\theta(x,r,\psi))\sum_{s\geq r}\{\alpha V_{e}(s,\psi) + (1 - \alpha)E_{z}[V_{e}(z,\psi) - x]\}f(s)
% \end{align}
\item Profits competed to zero (free entry):
  \begin{align}
    \hspace{-10mm}V_{v}(x,r,\psi) &= 0\nonumber\\
    \rightarrow\kappa &\geq q(\theta(x,r,\psi))\sum_{s\geq r}\{\alpha V_{e}(s,\psi) + (1 - \alpha)E_{z}[V_{e}(z,\psi) - x]\}f(s)
  \end{align}
\item Note, if $q^{-1}= \theta$ exists:
  \begin{align}
    \hspace{-10mm}q(\theta(x,r,\psi)) &= \frac{\kappa}{\sum_{s\geq r}\{\alpha V_{e}(s,\psi) + (1 - \alpha)E_{z}[V_{e}(z,\psi) - x]\}f(s)}\\
    \theta(x,r,\psi) &= q^{-1}(\frac{\kappa}{\sum_{s\geq r}\{\alpha V_{e}(s,\psi) + (1 - \alpha)E_{z}[V_{e}(z,\psi) - x]\}f(s)})
  \end{align}
\end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{Key Points}
\begin{itemize}
\item Free Entry in Random Search:
  \begin{align*}
    \hspace{-5mm}q(\theta) &= \frac{\kappa}{\underbrace{[\text{Expected surplus of match}]}_{Depends\;on\;distribution}[\underbrace{\text{Acceptance probability of match}}_{Depends\;on\;distribution}]}
  \end{align*}
\item Here:
  \begin{align*}
\hspace{-5mm}q(\theta(x,r,\psi)) &= \frac{\kappa}{\sum_{s\geq r}\{\alpha V_{e}(s,\psi) + (1 - \alpha)E_{z}[V_{e}(z,\psi) - x]\}f(s)}\\
    &= \frac{\kappa}{\underbrace{[\text{Expected surplus of match}]}_{Does\;not\;depend\;on\;distribution}}
  \end{align*}
\item Submarket indexed by value $x$ and reservation productivity $r$
\item $\rightarrow$ expected profits and $prob(s>=r)$ known.
\item Vacancy creation only depends on $\psi$ through $y$.
\end{itemize}
\end{frame}

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

\section{Equilibrium} % 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{Block Recursive Equilibrium}
A block-recursive equilibrium (BRE) consists of a market tightness function $\theta$, a value function for the unemployed worker $V_{u}$, a policy function for the unemployed worker $(x_{u}, r_{u})$, a joint value function for the firm-worker match $V_{e}$, and policy functions for the match $d$ and $(x_{e}, r_{e})$. These functions satisfy the following:
\begin{enumerate}
\item $\theta(x, r, y)$ satisfies free entry in all submarkets.
\item $V_{u}(y)$ satisfies the unemployed problem with associated policy functions $(x_{u}(y),r_{u}(y))$.
\item $V_{e}(z, y)$ satisfies the joint problem with associated policy functions $d(z, y)$ and $(x_{e}(z, y), r_{e}(z, y))$
\end{enumerate}\newline
\begin{enumerate}
\item[4'] The evolution of the aggregate state is consistent with all policy functions, $\psi' = \Psi(\psi)$.
\end{enumerate}
Block recursive means that the equilibrium solved without the last ``block'': 4' recovered via simulation.
\end{frame}

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

\begin{frame}
\frametitle{How does it work?}
\begin{itemize}
\item Simpler to see in a life-cycle model.
\item Matched value in terminal period ($T$):
  \begin{align*}
    V_{e}^{T}(z,\psi) &= y + z\\
    V_{e}^{T}(z,y) &= y + z
  \end{align*}
\item Free entry in terminal period:
  \begin{align*}
    \hspace{-5mm}q(\theta^{T}(x,r,\psi)) &= \frac{\kappa}{\sum_{s\geq r}\{\alpha V_{e}^{T}(s,\psi) + (1 - \alpha)E_{z}[V_{e}^{T}(z,\psi) - x]\}f(s)}\\
    \hspace{-5mm}q(\theta^{T}(x,r,y)) &= \frac{\kappa}{\sum_{s\geq r}\{\alpha V_{e}^{T}(s,y) + (1 - \alpha)E_{z}[V_{e}^{T}(z,y) - x]\}f(s)}
  \end{align*}
\end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{How does it work? (II)}
\begin{itemize}
\item Easy to show that $\psi = y$ for search \& unemp Bellman at $T$.
\item Matched value in $T-1$:
  \begin{align*}
    \hspace{-5mm}V_{e}^{T-1}(z,\psi) &= y + z + \beta E\big[\max_{d,x,r}\{d V_{u}^{T}(\hat{\psi}) \nonumber\\&+ (1 - d)[V_{e}^{T}(z,\hat{\psi}) + \lambda_{e}D^{T}(x, r, V(z,\hat{\psi}), \hat{\psi})]\}\big]\\
    \hspace{-5mm}V_{e}^{T-1}(z,y) &= y + z + \beta E\big[\max_{d,x,r}\{d V_{u}^{T}(\hat{y}) \nonumber\\&+ (1 - d)[V_{e}^{T}(z,\hat{y}) + \lambda_{e}D^{T}(x, r, V(z,\hat{y}), \hat{y})]\}\big]
  \end{align*}
\item This ``...is just as easy as solving the planner's problem in a representative agent model''
\end{itemize}
\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''}
\begin{itemize}
\item Calibration two models:
  \begin{enumerate}
  \item ``Experience'' good model ($\alpha = 0$)
  \item ``Inspection'' good model ($\alpha = 1$)
  \end{enumerate}
\item Weibull distribution for idiosyncratic productivity ($f(z)$)
\item Assume period length is 1 month.
\end{itemize}
\includegraphics[width=\textwidth]{./Tab2.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{Findings}
\begin{itemize}
\item Hit model economy with 1\% aggregate productivity increase.
\item Compare experience and inspection goods model.
\item Track:
  \begin{enumerate}
  \item Transition Rates (EU, UE, EE)
  \item Levels ($u, v, \theta$)
  \item Average Productivity
  \end{enumerate}
\item Compare volatility results with data (9,000 mos. sim.)
\item Find:
  \begin{enumerate}
  \item Experience goods model better at matching data.
  \item Both more accurate than canonical random search models.
  \item Still underpredict volatility.
  \end{enumerate}
\end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{Experience Model: Transition Rates}
\includegraphics[width=\textwidth]{./Fig2.png}
\end{frame}

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

\begin{frame}
\frametitle{Experience Model: Levels}
\includegraphics[width=\textwidth]{./Fig3.png}
\end{frame}

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

\begin{frame}
\frametitle{Experience Model: Productivity}
\includegraphics[width=\textwidth]{./Fig4.png}
\end{frame}

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

\begin{frame}
\frametitle{Experience Model: Volatility}
\begin{itemize}
\item Data:
\end{itemize}
\includegraphics[width=\textwidth]{./Tab1.png}\vspace{-5mm}
\begin{itemize}
\item Model:
\end{itemize}
\includegraphics[width=\textwidth]{./Tab3.png}
\end{frame}

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

\begin{frame}
\frametitle{Experience Model: Comparison w/ Random Search}
\begin{itemize}
\item Here:
\end{itemize}\vspace{-2mm}
\centering\includegraphics[width=0.85\textwidth]{./Tab3.png}\vspace{-2.75mm}
\begin{itemize}\vspace{0.5mm}
\item Canonical Models:
\end{itemize}
\centering\includegraphics[width=0.85\textwidth]{./Tab4.png}
\end{frame}

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

\begin{frame}
\frametitle{Inspection Model}
\begin{itemize}
\item Don't include table of results, but less successful:
\end{itemize}
\includegraphics[width=\textwidth]{./Fig5.png}
\end{frame}

%-----------------------------------------------_

\begin{frame}
\frametitle{Why is this useful?}
\begin{itemize}
\item Aggregate shocks often intractable in random OTJ search and matching framework.
  \begin{enumerate}
  \item Moscarini and Postel-Vinay (2009)
  \end{enumerate}
\item Heterogeneity hard to handle in random search framework.
\item Here: both much easier.
\begin{enumerate}
\item Menzio, Telyukova, and Visschers (2018): Life-cycle
\item Herkenhoff (multiple): risk aversion + housing delinquency, risk aversion + life-cycle + consumer credit \& default
\item Garriga and Hedlund (2018): risk aversion + mortgage debt
\end{enumerate}
\item Downsides:
  \begin{enumerate}
  \item Do workers reject job offers?
  \item Do some job postings have excess ``congestion''?
  \item What about realistic features like multiple applications?
  \end{enumerate}
\end{itemize}
\end{frame}

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

\section{Additional Applications}
% ------------------------------------------------

\begin{frame}
\frametitle{What does this mean more generally?}
\begin{itemize}
\item Consider a problem in which workers make the following decisions:
  \begin{enumerate}
  \item Decision over college attendance and non-defaultable student debt;
  \item Subsequent job search decision (on and off-the-job);
  \item Within-period unsecured borrowing and default;
  \item Human capital accumulation.
  \end{enumerate}
\item Generally very hard problem:
  \begin{itemize}
  \item Workers: integrate over distribution across all states to determine labor market.
  \item Firms: same.
  \end{itemize}
\item BRE: separate each market.
\end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Next Time}
  \begin{itemize}
  \item Last class!
  \item Review? Maybe my Affirmative Action paper.
  \item Don't forget your homework.
  \end{itemize}
\end{frame}


\end{document}
