%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Beamer Presentation
% LaTeX Template
% Version 1.0 (10/11/12)
%
% This template has been downloaded from:
% http://www.LaTeXTemplates.com
%
% License:
% CC BY-NC-SA 3.0 (http://creativecommons.org/licenses/by-nc-sa/3.0/)
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% ----------------------------------------------------------------------------------------
%	PACKAGES AND THEMES
% ----------------------------------------------------------------------------------------

\documentclass[hyperref={colorlinks=true}]{beamer}

\mode<presentation> {

  % The Beamer class comes with a number of default slide themes
  % which change the colors and layouts of slides. Below this is a list
  % of all the themes, uncomment each in turn to see what they look like.

  \usetheme{default}
  % \usetheme{AnnArbor}
  % \usetheme{Antibes}
  % \usetheme{Bergen}
  % \usetheme{Berkeley}
  % \usetheme{Berlin}
  % \usetheme{Boadilla}
  % \usetheme{CambridgeUS}
  % \usetheme{Copenhagen}
  % \usetheme{Darmstadt}
  % \usetheme{Dresden}
  % \usetheme{Frankfurt}
  % \usetheme{Goettingen}
  % \usetheme{Hannover}
  % \usetheme{Ilmenau}
  % \usetheme{JuanLesPins}
  % \usetheme{Luebeck}
  % \usetheme{Madrid}
  % \usetheme{Malmoe}
  % \usetheme{Marburg}
  % \usetheme{Montpellier}
  % \usetheme{PaloAlto}
  % \usetheme{Pittsburgh}
  % \usetheme{Rochester}
  % \usetheme{Singapore}
  % \usetheme{Szeged}
  % \usetheme{Warsaw}

  % As well as themes, the Beamer class has a number of color themes
  % for any slide theme. Uncomment each of these in turn to see how it
  % changes the colors of your current slide theme.

  % \usecolortheme{albatross}
  % \usecolortheme{beaver}
  % \usecolortheme{beetle}
  % \usecolortheme{crane}
  % \usecolortheme{dolphin}
  % \usecolortheme{dove}
  % \usecolortheme{fly}
  % \usecolortheme{lily}
  % \usecolortheme{orchid}
  \usecolortheme{rose}
  % \usecolortheme{seagull}
  % \usecolortheme{seahorse}
  % \usecolortheme{whale}
  % \usecolortheme{wolverine}

  % \setbeamertemplate{footline} % To remove the footer line in all slides uncomment this line
  % \setbeamertemplate{footline}[page number] % To replace the footer line in all slides with a simple slide count uncomment this line

  % \setbeamertemplate{navigation symbols}{} % To remove the navigation symbols from the bottom of all slides uncomment this line
}

\usepackage{graphicx} % Allows including images
\usepackage{booktabs} % Allows the use of \toprule, \midrule and \bottomrule in tables
\usepackage{cancel}
\usepackage{amsmath}
\usepackage{amssymb}
% \usepackage{showframe}
\usepackage{caption}

%\usepackage{subcaption}
\usepackage{tcolorbox}
\usepackage{tikz}
\usepackage{tabularx}
\usepackage{array}
\usepackage{pgfplots}
\tcbuselibrary{skins}


\usepackage{subfig}
\beamertemplatenavigationsymbolsempty
\usepackage{color, colortbl}
\definecolor{LRed}{rgb}{1,.8,.8}
\definecolor{MRed}{rgb}{1,.6,.6}

\usepackage{tikz}
\usetikzlibrary{shapes,arrows,shapes.multipart,fit,shapes.misc,positioning}
\newcommand{\der}[2]{\frac{\partial #1}{\partial #2}}
%\usepackage[labelformat=empty]{caption}

\newcommand{\ra}[1]{\renewcommand{\arraystretch}{#1}}

\newcommand\marktopleft[1]{%
    \tikz[overlay,remember picture]
        \node (marker-#1-a) at (0,1.5ex) {};%
}
\newcommand\markbottomright[1]{%
    \tikz[overlay,remember picture]
        \node (marker-#1-b) at (7ex,0) {};%
    \tikz[overlay,remember picture,thick,dashed,inner sep=3pt]
        \node[draw=black,rounded corners=0pt,fill=red,opacity=.2,fit=(marker-#1-a.center) (marker-#1-b.center)] {};%
}


% ----------------------------------------------------------------------------------------
%	TITLE PAGE
% ----------------------------------------------------------------------------------------

\title[]{Quantitative Macro-Labor:\\ Responding to Outside Offers with Sequential Auctions} % 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{Course 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: Allow firms to renegotiate wages rather than contract.
  \item Research proposal/Introduction due 10/1 (Tuesday after next).
  \item Outline of expectations:
    \begin{itemize}
    \item A (fairly) well-posed research question.
      \begin{itemize}
      \item Online document has lots of info on this.
      \item Don't worry too much about having the perfect question.
      \end{itemize}
    \item A discussion of your proposed empirical strategy:
      \begin{itemize}
      \item I estimate the effect of x on y and (hope to) find z.
      \item I use xxx data source.
      \end{itemize}
    \item A description of the mechanism you think explains this phenomenon.
      \begin{itemize}
      \item I show using a model that this is (hopefully) explained by xxxx.
      \item The key insight is that in the model, something interacts with something else and causes z.
      \end{itemize}
    \end{itemize}
    \item Presentations that week as well.
  \end{itemize}
\end{frame}

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

\section{Contracting Environments}
% ------------------------------------------------

\begin{frame}
  \frametitle{Contracting Environment in B-M Models}
  \begin{itemize}
  \item Standard Burdett-Mortensen
    \begin{itemize}
    \item Firms have homogeneous productivity.
    \item Cannot respond to outside offers.
    \item Contracts stipulate a permanent wage.
    \item Distribution of wages posted determined by eqm. wage posting game.
    \end{itemize}
  \item These contracts are suboptimal:
  \item Firm would like to retain workers, but artificially restricted:
    \begin{enumerate}
    \item Cannot respond to outside offers.
    \item Cannot change wage from first offered wage.
    \end{enumerate}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Contracting Environment in B-M Models}
  \begin{itemize}
  \item Burdett and Coles (2003):
    \begin{itemize}
    \item Firms have homogeneous productivity.
    \item Cannot respond to outside offers.
    \item {\it Contracts specify a value to be delivered over time in expectation}
    \item Distribution of determined by eqm. posting game.
    \end{itemize}
  \item These contracts are optimal given the environment:
    \begin{enumerate}
    \item Firm backloads contracts to reward workers for staying.
    \item Solves the ``moral hazard problem'' of on-the-job search.
    \end{enumerate}
  \end{itemize}
\end{frame}

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

% \begin{frame}
%   \frametitle{Burdett-Mortensen Models}
%   \begin{itemize}
%   \item Basic idea:
%     \begin{enumerate}
%     \item Workers can be in one of two states: employed or unemployed, with value functions $V, U$.
%     \item Firms post contracts, i.e., a given distribution of contracts, $V\in [\underline{V},\bar{V}], V\sim F(.)$.
%     \item Both employed and unemployed receive onctract offers at rates $(\lambda, \alpha)$, no prior info.
%     \item Separate two ways: exogenously (rate $\delta$) and via thru OTJS (rate $\lambda [1 - F(w)]$)
%     \item Discount rate $r$, generically utility $u(b), u(w)$.
%     \end{enumerate}
%   \item {\bf Firms cannot respond to outside offers.}
%   \item Equilibrium Objects:
%     \begin{enumerate}
%     \item Worker reservation strategies (on and off job).
%     \item Contracts posted by firms.
%     \item Worker distribution across employment states and contract values.
%     \end{enumerate}
%   \end{itemize}
% \end{frame}

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

% \begin{frame}
%   \frametitle{The Burdett-Mortensen Model}
%   \begin{itemize}
%   \item Flow value of unemployment:
%     \begin{align*}
%       rU &= u(b) + \alpha \int_{\underline{w}}^{\bar{w}}\max\{x - U, 0\}dF(x)
%     \end{align*}
%   \item Employment:
%     \begin{align*}
%       rV(t|w) &= u(w) + \lambda \int_{\underline{V}}^{\bar{V}}\max\{x - V(t|w), 0\}dF(x) + \delta (U - V(t|w))
%     \end{align*}
%   \item If wage contracts are fixed, i.e., $w_{0} = w_{t}\forall t$, and utility is linear:
%     \begin{align*}
%       w_{R} &= b + (\alpha - \lambda) \int_{w_{R}}^{\bar{w}}\frac{[1 - F(x)]}{r + \delta + \lambda[1 - F(w_{R})]}dx
%     \end{align*}
%   \end{itemize}
% \end{frame}

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

% \begin{frame}
%   \frametitle{The Firm}
%   \begin{itemize}
%   \item Assume equilibrium conditions \& $\lambda = \alpha$.
%   \item Define $\pi^{V}$ as the profits of a vacant firm.
%   \item Firm profit function:
%     \begin{align*}
%       \pi^{V} &= \max_{w} (p - w)l(w|w_{R}, F)\\
%       \pi^{V}(w|w_{R},F) &= (p - w)l(w|w_{R}, F)\\
%       l(w|w_{R}, F) &= \frac{m\alpha\delta}{(\delta + \alpha[1 - F(w)])^{2}}
%     \end{align*}
%   \item What is $l$? It is the probability of meeting a workers {\it and} the expected duration a worker employed at wage $w$ will be employed with a firm.
%   \end{itemize}
% \end{frame}

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

% \begin{frame}
%   \frametitle{Wage Contract}
%   \begin{itemize}
%   \item Solving the contracting problem yields the following contract:
%     \begin{align*}
%       \frac{u'(w^{*}(0|.))}{u'(w^{*}(\tau|.))} = 1 + u'(w^{*}(0|.))\int_{0}^{\tau}\lambda F'(V^{*}(t|.))\pi^{*}(t|.)dt
%     \end{align*}
%   \item Strictly increasing and concave utility:
%     \begin{align*}
%       \frac{u'(w^{*}(0|.))}{u'(w^{*}(\tau|.))} > 1 \rightarrow w^{*}(\tau) > w^{*}(0)
%     \end{align*}
%   \item Increasing wage at time $\tau$ decreases quit rate over $[0, \tau]$:
%     \begin{align*}
%       \underbrace{u'(w^{*}(0|.))}_{Slope}\int_{0}^{\tau}\underbrace{\lambda F'(V^{*}(t|.))}_{\Delta\;Quit\;Rate}\underbrace{\pi^{*}(t|.)}_{PDV\;of\;Profits}dt
%     \end{align*}
%   \item Moral hazard: reward those who do not quit.
%   \end{itemize}
% \end{frame}

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

\begin{frame}
  \frametitle{Empirical Regularities}
  \begin{itemize}
  \item We've primarily discussed the theory the last few weeks, but what are the predictions of these models?
  \item Burdett and Coles (2003):
    \begin{enumerate}
    \item Wage profiles are upward sloping.
    \item Wages increase when moving job-to-job.
    \item Job-to-job mobility slows as wages increase.
    \end{enumerate}
  \item What do we observe in the data? (some from Shouyong Shi's notes on directed search)
    \begin{enumerate}
    \item Wages increase with tenure (Farber, 99) $\checkmark$
    \item High wage workers less likely to quit (Farber, 99) $\checkmark$
    \item Dispersion among workers with identical tenure
    \item Workers moving {\it down} the wage ladder.
    \end{enumerate}
  \item Can we use an alternate contracting environment to explain the last two?
  \end{itemize}
\end{frame}

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

\section{Postel-Vinay and Robin (2002)}
% ------------------------------------------------

\begin{frame}
  \frametitle{Postel-Vinay and Robin (2002)}
  \begin{itemize}
  \item Now, a firm {\it can} respond to outside offers.
  \item Key ingredients:
    \begin{enumerate}
    \item Firm heterogeneity in terms of productivity.
    \item Fixed wage contracts.
    \end{enumerate}
  \item The contracts are fixed-wage, but can be {\it renegotiated}.
  \item Whenever a worker receives an offer, his current employer tries to convince him to stay.
  \item Current and offering firm have ``auction'' over worker (hence sequential auctions).
  \item Higher productivity firm wins.
  \item (Note: goal of paper is determining contribution of heterogeneity to wage dispersion, hence two-sided heterogeneity.)
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Environment}
  \begin{itemize}
  \item Agents:
    \begin{itemize}
    \item Workers are heterogeneous wrt employment status and ability (fixed).
    \item Worker ability: $\epsilon\sim H(.)$.
    \item Worker value functions: $V_{0}(\epsilon), V_{1}(\epsilon, w, p)$
    \item Firms are ex-ante heterogeneous wrt prod., $p\sim F(.), p\in [\underline{p},\bar{p}]$
    \end{itemize}
  \item Preferences and Technology:
    \begin{itemize}
    \item Production of a type-$(\epsilon, p)$ match: $\epsilon p$
    \item Unspecified utility: $u = U(\epsilon b)$, $u = U(w)$.
    \item Workers and firms meet at rate $\lambda_{0}$ (unemployed), $\lambda_{1}$ (employed).
    \item Exogenous separations, $\delta$, and birth/death $\mu$
    \end{itemize}
  \item Symmetric discount rate $\rho$.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Wage Determination}
  \begin{itemize}
  \item ``Sequential Auctions'' a poaching firm bids on a worker against his incumbent firm.
  \item Wage determination assumptions:
    \begin{enumerate}
    \item {\it Firms can vary their wage offers according to worker characteristics.}
    \item {\it They can counter offers made by competing firms.}
    \item All offers are take-it-or-leave-it.
    \item Contracts are long-term and can be renegotiated by mutual agreement.
    \end{enumerate}
  \item Take-it-or-leave-it offers are the result of game played between firms.
  \item This can generate {\it within-firm} variation in wages based on luck.
  \item Some workers happen to run into other firms more often $\rightarrow$ higher wages.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Unemployed Value Function}
  \begin{itemize}
  \item Unemployed flow value:
    \begin{align*}
      (\rho + \mu + \lambda_{0})V_{0}(\epsilon) &= U(\epsilon b) + \lambda_{0}\int_{p_{R}}^{\bar{p}}V(\epsilon, \phi_{0}(\epsilon, x), x)dF(x)
    \end{align*}
  \item What is $\phi_{0}(\epsilon, p)$? Function mapping $\phi_{0}: \bbm{R}_{\epsilon\times p}\rightarrow \bbm{R}_{+}$ heterogeneity to wages.
  \item Firms make take-it-or-leave-it offers.
  \item What is the
    \begin{enumerate}
    \item Wage offered to firms?
    \item Reservation ``mpl'' (they mean $p$)?
    \end{enumerate}
  \item What does take-it-or-leave-it offers mean about a worker's bargaining power?
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Employed Reservation Strategy}
\begin{center}
\begin{figure}[h]\label{fig:WealthLearningCorrelation}\centering
\begin{tikzpicture}
\begin{axis}[
    axis lines=both,
    % axis line style={--},
    x label style={at={(axis description cs:0.5,0)},anchor=north},
    y label style={at={(axis description cs:0.1,0.5)},rotate=0,anchor=south},
    % xtick=\empty, ytick=\empty,
    width=10cm, height=9cm,
    % ymajorgrids=true,
    % grid style=dashed,
    legend style={at={(0.8,0.3)},anchor=north,legend columns=1},
    legend cell align*={left},
    xtick pos = both, ytick pos = both,
    yticklabel pos=left,
    clip=false,
    xlabel = {Age},
    ylabel = {Labor Earnings (logs)},
    ymax = 12.5, ymin = 8,
    xmin = 23, xmax = 62
]

    \addlegendimage{mark=square*,blue}
    \addlegendentry{Low $p=10.5$ firm}
    \addlegendimage{mark=*,red}
    \addlegendentry{High $p=12$ firm}

%% earnings profile for top quintile
    \addplot[mark=square*, blue] table[row sep=\\]{
        X Y\\25 10.2\\30 10.3\\35 10.45\\40 10.5\\45 10.55\\50 10.6\\55 10.6\\60 10.6\\
    };

%% earnings profile for bottom quintile
    \addplot[mark=*, red] table[row sep=\\]{
        X Y\\25 9.5\\30 10\\35 10.5\\40 11\\45 11.5\\50 12\\55 12\\60 12\\
    };

% \node[anchor=west] (source) at (axis cs:22,11.8){\footnotesize \begin{tabular}{c} Slope of Wages \\ if $\sigma_{AL} \rightarrow 1$ \end{tabular}};
% \node[anchor=west] (sourceA) at (axis cs:36,11.7);
% \node[anchor=west] (sourceB) at (axis cs:36,11.6);
%        \node (destinationA) at (axis cs:43,11.2){};
%        \node (destinationB) at (axis cs:42,9.25){};
%        \draw[->](sourceA)--(destinationA);
%        \draw[->](sourceB)--(destinationB);
% \node[anchor=west] (source) at (axis cs:32,8.5){\footnotesize \begin{tabular}{c} Slope of Wages \\ if $\sigma_{AL} \rightarrow 0$ \end{tabular}};
% \node[anchor=west] (sourceA) at (axis cs:42,9);
% \node[anchor=west] (sourceB) at (axis cs:44,9);
%        \node (destinationA) at (axis cs:48,10.95){};
%        \node (destinationB) at (axis cs:48,9.85){};
%        \draw[->](sourceA) -- (destinationA);
%        \draw[->](sourceB) -- (destinationB);

\end{axis}
\end{tikzpicture}
\end{figure}
\end{center}
\end{frame}

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

\begin{frame}
  \frametitle{Equilibrium Wages}
  \begin{itemize}
  \item Worker with state $(\epsilon, w, p)$
  \item What is the maximum the incumbent firm $p$, could pay? $w = \epsilon p$.
  \item Worker could run into the following firms characterized by their productivity:
    \begin{enumerate}
    \item Firm $p' \leq \frac{w}{\epsilon}$:
      \begin{itemize}
      \item $p'$ so low that highest wage less than current wage. $\epsilon p') \leq w$
      \end{itemize}
    \item Firm $p' < p$, but $\epsilon p' > w$:
      \begin{itemize}
      \item $p'$ firm cannot outbid $p$ firm, but bids wage up.
      \end{itemize}
    \item Firm $p' > p$:
      \begin{itemize}
      \item Incumbent firm cannot match poaching firm. Wage falls to compensate poaching firm for future wage increases.
      \end{itemize}
  \end{enumerate}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Equilibrium Wages}
  \begin{itemize}
  \item $\phi$: wage that makes worker indifferent given $\epsilon$ and productivities $p$, $p'$. Second argument is always $\tilde{p}>\hat{p}$.
  \item Define a productivity threshold $q$ such that
    \begin{align*}
      \phi(\epsilon, q(\epsilon,w,p), p) = w
    \end{align*}
  \item $q$ is the lowest productivity firm $p\in [\underline{p},\bar{p}]$ from which an offer can impact the wage.
  \item Corresponding continuation values and probabilities:
    \begin{enumerate}
    \item Firm $p' \leq \frac{w}{\epsilon}$:
      \begin{itemize}
      \item Probability: $F(q(\epsilon, w, p))$, CV: $V(\epsilon, w, p)$.
      \end{itemize}
    \item Firm $p' < p$, but $\phi(\epsilon, p', p) > w$:
      \begin{itemize}
      \item $F(p) - F(q)$, $V_{t + 1} = V(\epsilon, \phi(\epsilon, p', p), p) = V(\epsilon, \epsilon p', p')$
      \end{itemize}
    \item Firm $p' > p$:
      \begin{itemize}
      \item $1 - F(p)$, $V_{t + 1} = V(\epsilon, \phi(\epsilon, p, p'), p') = V(\epsilon, \epsilon p, p)$
      \end{itemize}
    \end{enumerate}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Wage Cuts while Moving up Ladder}
  \begin{itemize}
  \item As an example, consider two firms with income growth rates $\gamma_{1}$ and $\gamma_{2}$, $\gamma_{2} > \gamma_{1}$.
  \item You are currently employed by firm 1 at a wage $y_{1}$, and firm 2 is offering you $y_{2}$.
  \item You must work for whoever you pick permanently, and you are maximizing lifetime income with discount rate $\beta$.
  \item Lifetime income:
    \begin{align*}
      \sum_{t = 0}^{\infty}((1 + \gamma_{j})\beta)^{t}y_{j}
    \end{align*}
  \item Present values:
    \begin{enumerate}
    \item Firm 1: $\frac{y_{1}}{1 - \beta(1 + \gamma_{1})}$
    \item Firm 2: $\frac{y_{2}}{1 - \beta(1 + \gamma_{2})}$
    \end{enumerate}
  \item In this case, what we are saying is that firm 2 would pick $y_{2}$ st
    \begin{align*}
      y_{2} = \frac{y_{1}(1 - \beta(1 + \gamma_{2}))}{1 - \beta(1 + \gamma_{1})}
    \end{align*}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Employed Value Function}
  \begin{itemize}
  \item Flow value of employment ($q = q(\epsilon, w, p)$):
    \begin{align*}
      (\rho + \delta + \mu)V_{1}(\epsilon, w, p) &= U(w) + \delta V_{0}(\epsilon) \nonumber\\&+ \lambda_{1}\int_{q}^{p}V(\epsilon, \phi(\epsilon, p, x), p)dF(x)\\
      (\rho + \delta + \mu)V_{1}(\epsilon, w, p) &= U(w) + \delta V_{0}(\epsilon) \nonumber\\&+ \lambda_{1}\int_{q}^{p}[1 - F(x)]\der{V}{\phi}\der{\phi}{x}dx
    \end{align*}
  \item How do we find $\der{V}{\phi}\der{\phi}{x}$? From $q$ and $p$, any competing offer $\rightarrow V(\epsilon, \phi, p) = V(\epsilon, \epsilon x, x)$.
    \begin{align*}
      \rightarrow V(\epsilon, \epsilon p, p) &= \frac{U(\epsilon p) + \delta V_{0}(\epsilon)}{\rho + \delta + \mu}
    \end{align*}
  \end{itemize}
  \begin{align*}\hspace{-8mm}
\rightarrow (\rho + \delta + \mu)V_{1}(\epsilon, w, p) &= U(w) + \delta V_{0}(\epsilon) \nonumber\\&+ \frac{\lambda_{1}\epsilon}{\rho + \delta + \mu}\int_{q}^{p}[1 - F(x)]U'(x)dx
  \end{align*}
\end{frame}

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

\begin{frame}
  \frametitle{Reservation Strategies}
  \begin{itemize}
  \item Employed reservation strategy:
    \begin{align*}
      V(\epsilon, \phi(\epsilon, p, p'), p') &= V(\epsilon, \epsilon p, p)\\
      V(\epsilon, \phi(\epsilon, p, p'), p') - V(\epsilon, \epsilon p, p) &= 0\\
      \rightarrow V(\epsilon, \phi, p') - \frac{U(\epsilon p) + \delta V_{0}(\epsilon)}{\rho + \delta + \mu}&= 0
    \end{align*} \\
      \item From earlier: $V(\epsilon, \epsilon p, p) = \frac{U(\epsilon p) + \delta V_{0}(\epsilon)}{\rho + \delta + \mu}$.
  \begin{align*}
\underbrace{U(\phi(\epsilon,p,p'))}_{Poaching\:Utility} &= \underbrace{U(\epsilon p)}_{Incumbent\;Utility} \nonumber\\&- \underbrace{\frac{\lambda_{1}}{\rho + \delta + \mu}}_{Offer\;Arrival}\underbrace{\int_{p}^{p'}[1 - F(x)]\epsilon U'(\epsilon x)dx}_{Wage\;Growth\;Utility}
  \end{align*}
\item Inverting this function yields the reservation strategies.
\item Identical argument for unemployed workers.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Decomposition}
  \begin{itemize}
  \item Conveniently, reservation equation log-linearizes for different utility functions (CRRA, $U(c) = \frac{c^{1 - \alpha}-1}{1 - \alpha}$):
    \begin{align*}
      ln(\phi(&\epsilon,p,p')) = ln(\epsilon) + ln(\phi(1,p,p'))\\
      ln(\phi(&\epsilon,p,p')) = ln(\epsilon) \nonumber\\&+ \frac{1}{1 - \alpha}ln(p^{1 - \alpha} - \frac{\lambda_{1}(1 - \alpha)}{\rho + \delta + \mu}\int_{p}^{p'}[1 - F(x)]x^{-\alpha}dx),\hfill \alpha\cancel{=}1\\
      ln(\phi(&\epsilon,p,p')) = ln(\epsilon) + ln(p) \nonumber\\&- \frac{\lambda_{1}}{\rho + \delta + \mu}\int_{p}^{p'}[1 - F(x)]\frac{dx}{x},\hfill \alpha=1
    \end{align*}
  \item Here, $ln(\epsilon)$ is the {\it worker} effect.
  \item And $ln(\phi(1,p,p'))$ is the labor market history effect.
  \end{itemize}
\end{frame}

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

\section{Equilibrium}

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

\begin{frame}
  \frametitle{Steady-State Equilibrium}
  \begin{itemize}
  \item They are interested in the cross sectional dispersion of wages, so they focus on the steady-state.
  \item ``The steady state assumption implies that inflows must balance outflows for all stocks of workers defined by a status (unemployed or employed), a personal type $\epsilon$, a wage $w$, and an employer type $p$.''
  \item The equilibrium objects are
    \begin{enumerate}
    \item Reservation strategies for each worker over firm productivities, given the distributions and prices.
    \item Wage function for for each tuple $(\epsilon, p, p')$ with $p' = b$ for unemployed, given the distributions.
    \item Flow equations that balance according to the statement above.
    \end{enumerate}
  \item They derive the distributions in the paper.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Log-Wage Variance}
  \begin{itemize}
  \item We will define a firm by its productivity ``type''
  \item Recall definition of conditional variance:
    \begin{align*}
      V(x) = E[V(x|y)] + V[E(x|y)]
    \end{align*}
  \item The log-linearity of wages is very useful!
    \begin{align*}
      ln(\phi(&\epsilon,q,p)) = ln(\epsilon) + ln(\phi(1,q,p))\\
      \rightarrow E[ln(\phi(&\epsilon,q,p))|p] = E[ln(\epsilon)] + E[ln(\phi(1,q,p))|p]\\
      \rightarrow V[ln(\phi(&\epsilon,q,p))|p] = V[ln(\epsilon)] + V[ln(\phi(1,q,p))|p]
    \end{align*}
  \item Then the total variance of wages is given by
    \begin{align*}\small\hspace{-5mm}
      V(ln(w)) &= V(ln(\epsilon)) + V (E[ln(w|p)]) + (E[V(ln(w|p))] - V(ln(\epsilon)))\\
               &= \underbrace{V(ln(\epsilon))}_{Individual} + \underbrace{V (E[ln(\phi(1,q,p))|p])}_{Between\;Firm} \\&+ \underbrace{E[V(ln(\phi(1,q,p))|p)]}_{Within\;Firm\;non-individual}
    \end{align*}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Empirical Analysis}
  \begin{itemize}
  \item They use a matched employer-employee dataset from France.
  \item They estimate the model, and then use simulated data to decompose the size of the worker effect, the firm effect, and the labor market effect.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Decomposition by Occupation (Postel-Vinay and Robin, 2002)}
\includegraphics[width=1.1\textwidth]{Postel-VinayRobin1.png}
\end{frame}

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

% \begin{frame}
%   \frametitle{Confidence Intervals}
% \includegraphics[width=1.1\textwidth]{Postel-VinayRobin4.png}
% \end{frame}

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

\begin{frame}
  \frametitle{Job-Stayers Wage Growth (yearly, Postel-Vinay and Robin, 2002)}
\includegraphics[width=1.1\textwidth]{Postel-VinayRobin3.png}
\end{frame}

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

\begin{frame}
  \frametitle{Job-to-Job Wage Growth (yearly, Postel-Vinay and Robin, 2002)}
\includegraphics[width=1.1\textwidth]{Postel-VinayRobin2.png}
\end{frame}

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

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

\begin{frame}
  \frametitle{Next Time}
  \begin{itemize}
  \item Thursday: Equilibrium search and matching: Mortensen-Pissarides.
  \item Next Tuesday: presentations of your research proposal/introduction
  \end{itemize}
\end{frame}


\end{document}