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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{Announcements}
  \begin{itemize}
  \item Today:
    \begin{enumerate}
    \item Random vs. directed search
    \item Change primitive: workers can observe wage offers prior to match.
    \item i.e., Moen (1997)
    \end{enumerate}
  \item Homework due on Wednesday.
  \item Final: 24 hours starting Friday at 9am.
  \item 3 questions, any part of course is fair game.
  \end{itemize}
\end{frame}

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


\begin{frame}
  \frametitle{Random vs. Directed Search}
  \begin{itemize}
  \item How do workers find jobs?
  \item How much information do they have about a job before applying?
  \item Two extremes:
    \begin{enumerate}
    \item Random Search: {\it no} information about a job prior to receiving offer.
    \item Directed Search: {\it all} information about a job prior to application.
    \end{enumerate}
  \item Why does this matter?
    \begin{enumerate}
    \item Random search is generically inefficient: one worker may reject a job offer than another would accept.
    \item Directed search is generically efficient: by applying for a job, a worker signals that the job is already acceptable.
    \end{enumerate}
  \item As we will see next time, it also changes computational complexity.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Random vs. Directed Search II}
  \begin{itemize}
  \item Empirically, how can we tell them apart?
  \item Hazard rate to wage w generically:
    \begin{align}
      H_{U}(w) = \underbrace{\lambda(w)}_{Arrival\;Rate} \times \underbrace{f(w)}_{Prob. Offer\;=\;W}
    \end{align}
  \item Random search:
    \begin{align}
      H_{U}(w) = \underbrace{\lambda}_{Arrival\;Rate}\underbrace{[1 - F(w_{R})]}_{Selectivity}\underbrace{f(w)}_{P.(Offer\;=\;W)}
    \end{align}
  \item Directed search:
    \begin{align}
      H_{U}(w) &= \underbrace{\lambda(w)}_{Arrival}\underbrace{\cancel{[1 - F(w_{R})]}}_{F(w_{R})=0}\underbrace{f(w)}_{P(Offer\;=\;w)}\\
               &= \underbrace{\lambda(W)}_{Arrival}\cancel{\underbrace{f(w)}_{P(w=w_{j})=1}}\\
               &= \underbrace{\lambda(w)}_{Arrival\;Rate\;\;of\;Wage\;w}
    \end{align}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Some Evidence}
  \begin{itemize}
  \item (First couple Borrowed from Shouyong Shi)
  \item Hall and Krueger (08):
    \begin{enumerate}
    \item 84\% had information on wage prior to first interview.
    \end{enumerate}
  \item Holzer, Katz, and Krueger (91)
    \begin{enumerate}
    \item Firms in high-wage industries receive more applications than low-wage industries, controlling for observables.
    \end{enumerate}
  \item Braun, Engelhardt, Griffy, and Rupert: unemployment insurance changes $\lambda(w)$ $\rightarrow$ inconsistent with random search.
  \end{itemize}
\end{frame}

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

\section{The Mortensen and Pissarides Model}
% ------------------------------------------------

\begin{frame}
  \frametitle{Mortensen and Pissarides Model}
  \begin{itemize}
  \item Unemployed flow value:
    \begin{align}
      rU = b + p(\theta)E[W(w) - U]
    \end{align}
  \item Employed flow value:
    \begin{align}
      rW(w) = w + \delta[U - W(w)]
    \end{align}
  \item Vacant flow value:
    \begin{align}
      rV = -\kappa + q(\theta)E[J(w) - V]
    \end{align}
  \item Matched flow value:
    \begin{align}
      rJ(w) = (p - w) + \delta[V - J(w)]
    \end{align}
  \item Free entry equilibrium condition:
    \begin{align}
      V &= 0\\
      \rightarrow \frac{\kappa}{E[J(w)]} &= q(\theta)
    \end{align}
  \end{itemize}
\end{frame}

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


\begin{frame}
  \frametitle{Equilibrium}
  \begin{itemize}
  \item The equilibrium we have described is a steady-state equilibrium characterized by value functions $U, W, J, V$, a wage function $w$, a market tightness function $\theta$, and steady-state level unemployment $u$, such that
    \begin{enumerate}
    \item A steady-state level of unemployment, derived from the flow unemployment equation.
    \item A wage rule that splits the surplus of a match according to a sharing rule with bargaining weight $\beta$
    \item A free entry condition that determines $\theta$ given wages and steady-state unemployment.
    \end{enumerate}
  \item What were these policy functions?
    \begin{enumerate}
    \item $w = (1 - \beta) b + \beta p + \beta\theta\kappa$
    \item $\theta = q^{-1}(\frac{\kappa(r + \delta)}{(p - w)})$
    \item $u &= \frac{\delta}{\delta + p(\theta)}$
    \end{enumerate}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Equilibrium}\label{surplus}
\centering\begin{tikzpicture}[scale=0.8]


% Axis

\draw [thick] (-0.3,0) node [below] (-0.5,0)-- (0,0) -- (5.5,0) node [right];

\node [below] at (5.5,-0.2) {$w^{*}$};

\draw [thick] (0,-0.5)-- (0,0) -- (0,5.5);

\node [left] at (0,5.3) {$\theta$};


%Downward slopping line

\node [above] at (0.5,5);

\draw [thick] (0.5,5) to [out=280,in=140] (4.5,0);

\node [above] at (4.5,0.2) {$VCC$};


% Upward Slopping PQ

\draw [thick] (0.3,0.5) -- (3.8,4);

\node [right] at (3.8,4) {$w(\theta)$};


% dashed lines

\draw [dashed](2,0)--(2,2.15);

\node [below] at (2,0) {$w^{*}$};

\draw [dashed](0,2.15)--(5,2.15);

\node [left] at (0,2.15) {$\theta^{*}$};

\end{tikzpicture}%
\begin{tikzpicture}[scale=0.8]


% Axis

\draw [thick] (-0.3,0) node [below] (-0.5,0)-- (0,0) -- (5.5,0) node [right];

\node [below] at (5.5,-0.2) {$u$};

\draw [thick] (0,-0.5)-- (0,0) -- (0,5.5);

\node [left] at (0,5.3) {$\theta$};


%Downward slopping line

\node [above] at (0.5,5) {$BC$};

\draw [thick] (0.5,5) to [out=280,in=140] (4.5,0);


% dashed lines

\draw [dashed](2,0)--(2,2.15);

\node [below] at (2,0) {$u^{*}$};

\draw [dashed](0,2.15)--(2,2.15);

\node [left] at (0,2) {$\theta^{*}$};

\end{tikzpicture}
\end{frame}

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

\section{Directed Search}
% ------------------------------------------------

\begin{frame}
  \frametitle{Directed/Competitive Search}
  \begin{itemize}
  \item In DMP, wages are negotiated/revealed after meeting.
  \item This can create inefficiency:
    \begin{enumerate}
    \item Consider example with unemployed workers A and B.
    \item $w_{R}^{A} = 10$, $w_{R}^{B} = 15$
    \item Firm pays a cost $\kappa$ to open a vacancy and posts a wage 12.
    \item Both worker A and B apply for the job. Firm randomly picks worker B.
    \item Worker B rejects job that would have been acceptable to worker A.
    \end{enumerate}
  \item Directed search: Worker B applies for different job with $w\geq w_{R}^{B}$.
  \item (Directed and competitive search generally used interchangeably).
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{The Competitive Search Model (Moen, 1997)}
  \begin{itemize}
  \item Agents:
    \begin{enumerate}
    \item Employed workers employed in submarket $i$;
    \item unemployed workers considering searching in $i\in\{1,...,N\}$;
    \item unmatched firms indexed by productivity $y_{i}\in{y_{1},...,y_{N}}$;
    \item matched firms indexed by productivity $y_{i}\in{y_{1},...,y_{N}}$;
    \item ``Market Maker'': benevolent overlord who announces eqm. $w_{i}$.
    \end{enumerate}
  \item Linearity: ($u = z, u = w_{i}$) and $y = y_{i}>z$ in open submarkets.
  \item Matching function:
    \begin{enumerate}
    \item Determines {\it number} of meetings between firms \& workers in submarket $i$:
    \begin{align}
      M(u_{i}L_{i},v_{i}L_{i}) = u_{i}L_{i}\times M(1,\frac{v_{i}}{u_{i}}) = u_{i}L_{i}\times p(\theta_{i})
    \end{align}
  \item where $\theta_{i} = \frac{v_{i}}{u_{i}}$ is ``submarket tightness''
  \item Match rates:
    \begin{align}
      \underbrace{p(\theta_{i})}_{Worker\;wage\;i}= \theta_{i} \underbrace{q(\theta_{i})}_{Firm\;wage\;i}
    \end{align}
    \end{enumerate}
  \item $i$ indexes both the productivity and wage.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Worker Value Functions}
  \begin{itemize}
  \item Value functions:
    \begin{enumerate}
    \item Employed in submarket $i$: $W_{i}$
    \item Unemployed and searching in submarket $i$: $U_{i}$.
    \item Unemployed: $U = \max\{U_{1},...,U_{N}\}$.
    \end{enumerate}
  \item Unemployed flow value in submarket $i$:
    \begin{align}
      rU_{i} = z + p(\theta_{i})(W_{i} - U_{i})
    \end{align}
  \item Employed flow value in submarket $i$:
    \begin{align}
      rW_{i} = w_{i} + \delta(U_{i} - W_{i})
    \end{align}
  \item Both problems are stationary: optimal choice of $i$ true $\forall\;t$.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Worker Value Functions II}
  \begin{itemize}
  \item We can solve for match rates:
    \begin{align}
      rU_{i} &= z + p(\theta_{i})(W_{i} - U_{i})\\
      (r + p(\theta_{i}))U_{i} &= z + p(\theta_{i})\frac{w_{i} + \delta U_{i}}{r + \delta}\\
      (r + \delta)(r + p(\theta_{i}))U_{i} - p(\theta_{i})\delta U_{i} &= (r + \delta)z + p(\theta_{i})w_{i} \\
      rU_{i} &= \frac{(r + \delta)z + p(\theta_{i})w_{i}}{(r + \delta + p(\theta_{i}))} \\
      p(\theta_{i}) &= \frac{rU_{i} - z}{w_{i} - rU_{i}}(r + \delta) \\
    \end{align}
  \item $U = \max\{U_{1},...,U_{N}\}$ and ex-ante homogeneity among workers implies
    \begin{align}
      p(\theta_{i}) &= \frac{rU - z}{w_{i} - rU}(r + \delta) \\
    \end{align}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Firm Value Functions}
  \begin{itemize}
  \item Pays a cost $\chi$ to draw productivity.
  \item Firm observes own productivity, chooses to open vacancy given submarkets $(w, \theta)$.
  \item Value functions:
    \begin{enumerate}
    \item Vacant with productivity $y_{i}$: $V(y_{i},w,\theta)$
    \item Filled with productivity $y_{i}$, paying wage w: $J(y_{i}, w)$
    \end{enumerate}
  \item Vacant flow value:
    \begin{align}
      rV(y_{i},w,\theta) = -\kappa + q(\theta)(J(y_{i},w) - V(y_{i},w,\theta))
    \end{align}
  \item Matched flow value:
    \begin{align}
      rJ(y_{i},w) = y_{i} - w + \delta(V(y_{i},w,\theta) - J(y_{i},w))
    \end{align}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Firm Value Functions II}
  \begin{itemize}
  \item Value functions:
    \begin{enumerate}
    \item Vacant with productivity $y_{i}$: $V(y_{i},w,\theta)$
    \item Filled with productivity $y_{i}$, paying wage w: $J(y_{i}, w)$
    \end{enumerate}
  \item In equilibrium $V(y_{i},w,\theta) = 0$:
    \begin{align}
      rJ(y_{i},w) = y_{i} - w - \delta J(y_{i},w)
    \end{align}
  \item Asset value of vacancy in submarket $(y_{i},w,\theta)$:
    \begin{align}
      (r + q(\theta))V(y_{i},w,\theta) &= q(\theta)\frac{y_{i} - w}{r + \delta} - \kappa
    \end{align}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Equilibrium}
  \begin{itemize}
  \item We will be interested in the same equilibrium objects, but now for each submarket $i$:
    \begin{enumerate}
    \item Wages $w_{i}$;
    \item unemployment $u_{i}$;
    \item $\theta_{i} = \frac{v_{i}}{u_{i}}$ vacancies in each submarket.
    \end{enumerate}
  \item Before, 1 \& 3 were separate equilibrium conditions.
  \item New equilibrium objects
    \begin{enumerate}
    \item set of open submarkets, $\mathcal{I}$;
    \item value of unemployment $\bar{V}(U)$
\end{enumerate}
  \item Market maker sets wages according to
    \begin{align}
      \max_{w} V(y_{i},w,\theta(w;U))
    \end{align}
  \item Given $p(\theta)$ from worker's problem, find $w$ that maximizes value of vacancy.
  \end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{Free Entry}

\begin{itemize}
	\vfill \item Free entry implies that the expected value of opening a vacancy will be equal to the cost of opening it $\chi$
	\vfill \item The expected value of opening a vacancy
	$$\bar{V}(U) = \sum_{i=\iota(U)}^{n} f_i V(y_i,w_i^*(U),\theta_i^*(U))$$
	\vfill \item equilibrium:
	$$\bar{V}(U) = \kappa$$
\end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{Equilibrium Number of Markets}

\begin{itemize}
	\vfill \item We know that each productivity will form a separate market
	\vfill \item There are $n$ productivities in the distribution
	\vfill \item All submarkets such that $w_i \ge rU$ will remain open
	\vfill \item Let $\iota$ denote the lowest submarket open
\end{itemize}

\end{frame}

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

\begin{frame}
  \frametitle{``Competitive'' Search}
  \begin{itemize}
  \item What is shaded region?
  \end{itemize}
\centering\includegraphics[width=0.8\textwidth]{Moen1.png}
\end{frame}

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

\begin{frame}
  \frametitle{``Competitive'' Search}
  \begin{itemize}
  \item Inefficiency (rejected matches in DMP)
  \end{itemize}
\centering\includegraphics[width=0.8\textwidth]{Moen2.png}
\end{frame}

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


\begin{frame}
  \frametitle{Equilibrium}
  \begin{itemize}
  \item The resulting competitive equilibrium with frictional labor markets is characterized by the following equations
    \begin{align}
      \bar{V}(U) &= \chi\\
      w_{i} &= \arg\max V(y_{i},w,\theta(w;U)), i\geq i_{R}\\
      rU_{i} &= \frac{(r + \delta)z + p(\theta_{i})w_{i}}{(r + \delta + p(\theta_{i}))}, i\geq i_{R}\\
      \dot{u}_{i} &= 0; u_{i}p(\theta_{i}) = e_{i}\delta\\
      \sum_{i}u_{i} &= u
    \end{align}
  \end{itemize}
\end{frame}

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

% \begin{frame}
%   \frametitle{Wage Posting}
%   \begin{itemize}
%   \item In previous description, wages were ``announced in equilibrium'' by a market maker.
%   \item Would firms choose to deviate if they set their own wages?
%   \item Suppose firms deviate and offer $w'$:
%   \end{itemize}
% \centering\includegraphics[width=0.7\textwidth]{Moen3.png}
% \end{frame}

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

% \begin{frame}
%   \frametitle{Wage Dispersion}
%   \begin{itemize}
%   \item Absent any ex ante heterogeneity on the worker side, is there still wage dispersion?
%   \item Workers indifferent between open submarkets in equilibrium:
%   \end{itemize}
% \centering\includegraphics[width=0.7\textwidth]{Moen4.png}
% \end{frame}

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

\begin{frame}
  \frametitle{What is ``valuable'' about directed search?}
  \begin{itemize}
  \item Submarkets are individually priced.
  \item i.e., contracts $(w,\theta)$ are known given the state of the worker and firm.
  \item Then, assuming that free entry binds in every open submarket, no longer need to condition on aggregate distributions as state variables (Menzio and Shi, 2011).
  \item So models are computationally tractable.
  \item Makes it possible to easily incorporate heterogeneity.
  \end{itemize}
\end{frame}

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

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

\begin{frame}
  \frametitle{Conclusion}
  \begin{itemize}
  \item Thank you for a good semester!
  \item Hope everyone has a good summer and please come visit once we can all be in person.
  \item Final on Friday, 24 hours to take exam, 3 hours to complete once started.
  \item 3 questions, anything is fair game.
  \end{itemize}
\end{frame}


\end{document}
