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% ----------------------------------------------------------------------------------------
%	TITLE PAGE
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\title[]{Quantitative Macro-Labor:\\ Wage Dispersion and Frictional Labor Markets} % 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 I posted instructions for the introduction/research proposal on my website.
  \item There should be code on the cluster to work with different models and different datasets.
  \item Can everyone
    \begin{enumerate}
    \item access campus cluster?
    \item install Stata?
    \end{enumerate}
  \end{itemize}
\end{frame}

% ------------------------------------------------
\section{Market Imperfections} % 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{Why are Similar Workers Paid Differently?}
  \begin{itemize}
  \item Posed by Dale Mortensen in his book ``Wage Dispersion''
  \item Abowd, Kramarz, and Margolis (1999): ``That... observably equivalent individuals earn markedly different compensation and have markedly different employment histories--is one of the enduring features of empirical analyses of labor markets...''
  \item What are some possible reasons?
    \begin{enumerate}
    \item Ability
    \item Selectivity
    \end{enumerate}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Residual Wage Dispersion}
  \begin{itemize}
  \item We will look to theory to understand {\it residual wage dispersion}: wage/income dispersion left over after we condition on observables.
  \item There's a lot:
    \begin{enumerate}
    \item Mortensen (2005): 70\% of wage dispersion is unexplained.
    \end{enumerate}
  \item Understanding where this comes from is (one of) the goal of labor economics.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Unconditional Wage Dispersion across Industries}
\centering\includegraphics[width=0.8\textwidth]{OiIdson1.png}
\end{frame}

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

\begin{frame}
  \frametitle{Unexplained Variation}
\centering\includegraphics[width=0.8\textwidth]{OiIdson2.png}
\end{frame}

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

\begin{frame}
  \frametitle{Abowd, Kramarz, and Margolis (1999)}
  \begin{itemize}
  \item Famous paper for estimating the size of worker and firm effects on residual wage dispersion.
  \item Longitudinal panel of matched employer-employee observations in France (insert joke about France here).
  \item Empirical specification:
    \begin{align}
      ln(y_{it}) &= \mu_{y} + \theta_{i} + \psi_{j,t} + (x_{it} - \mu_{x})\beta + \epsilon_{it}\\
      y_{it} &: income\\
      \mu_{y} &: average\;income\;in\;year\;t\\
      \theta_{i}&: individual\;FEs\\
      \psi_{j,t}&: firm\;FEs
    \end{align}
  \item Key findings:
    \begin{enumerate}
    \item Individual FEs explain more than Firm FEs.
    \item Ind. FEs: 90\% of inter-industry wage differentials.
    \item 75\% of the firm-size wage differentials.
    \end{enumerate}
  \end{itemize}
\end{frame}

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



\begin{frame}
  \frametitle{Abowd, Kramarz, and Margolis (1999)}
  \begin{itemize}
  \item Ind. FEs ($\theta$) strongly correlated with income ($y$), Firm FEs ($\psi$) not as much.
  \end{itemize}
\hspace{-10mm}\includegraphics[width=1.18\textwidth]{AbowdEtAl4.png}
\end{frame}

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


% \begin{frame}
%   \frametitle{Abowd, Kramarz, and Margolis (1999)}
% \begin{center}
% \includegraphics[width=0.9\textwidth]{AbowdEtAl2.png}
% \end{center}
%   \begin{itemize}
%   \item Ind. FE strongly correlated with industry FEs.
%   \end{itemize}
% \end{frame}

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

% \begin{frame}
%   \frametitle{Abowd, Kramarz, and Margolis (1999)}
% \begin{center}
% \includegraphics[width=0.8\textwidth]{AbowdEtAl3.png}
% \end{center}
%   \begin{itemize}
%   \item Firm FEs not so much.
%   \end{itemize}
% \end{frame}

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


\begin{frame}
  \frametitle{Abowd, Kramarz, and Margolis (1999)}
  \begin{itemize}
  \item These are estimates of the size of firm and worker effects.
  \item But they are still {\it reduced-form}.
  \item We haven't identified the underlying causes of the size of each.
  \item What are some possible heterogeneities among workers?
  \item What are some possible heterogeneities among firms and industries?
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Other Interesting Regularities}
  \begin{itemize}
  \item Davis and Haltiwanger (1991, 1996) on the level and growth in wage-size effects and wage dispersion between plants:
    \begin{enumerate}
    \item Plants with $> 5,000$ employees: \$3.14/hour more than plants with 25-49 in 1967.
    \item Between 1967 and 1986, real wage grew by \$1.00, but differential grew to \$6.31.
    \item Explains 40\% of the between-plant wage dispersion.
    \item between-plant accounts for 59\% of the total variance; within-plant accounts for 2\%.
    \item Mean wage grows as plant size grows; wage dispersion falls!
    \end{enumerate}
  \item So is there wage dispersion in the economy?
  \item Why?
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Perfectly Competitive Labor Markets}
  \begin{itemize}
  \item We typically think of markets as being perfectly competitive/walrasian, etc.
  \item Prices are determined by the point where supply = demand, and there is no excess.
  \item Implications for labor market:
    \begin{enumerate}
    \item Workers are paid $w = F_{L}(K, L)$, i.e., their marginal product.
    \item Zero profits in equilibrium.
    \end{enumerate}
  \item Wage dispersion {\it can} exist:
    \begin{enumerate}
    \item Dispersion directly proportional to dispersion in productivity/ability/human capital, etc.
    \end{enumerate}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Frictional Labor Market}
  \begin{itemize}
  \item But perfect competition is an approximation, both for analytical and computational simplicity.
  \item Things we observe:
    \begin{enumerate}
    \item Price dispersion among identical workers/goods.
    \item Failure of markets to clear: unemployment.
    \item Profits.
    \end{enumerate}
  \item Market imperfections (frictions): agents are profit maximizing, but lack of information and randomness prevent markets from perfectly clearing.
  \item $w \cancel{=} F_{L}(K, L)$.
  \item Here: explore job search as explanation for (some) wage dispersion.
  \end{itemize}
\end{frame}

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

\section{The McCall 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{Outline: Frictional Labor Markets}
  \begin{itemize}
  \item We'll explore the following:
    \begin{enumerate}
    \item Partial equilibrium job search models: there is some wage distribution and workers optimize by specifying a reservation threshold.
    \item Extending the partial equilibrium model: on-the-job search, wage-tenure contracts, risk-aversion.
    \item General equilibrium job search: introduce an entry decision on the firm's side and endogenize the matching rate.
    \item Efficiency and Directed search.
    \end{enumerate}
  \item Failings of the search framework:
    \begin{enumerate}
    \item Shimer (2005): can't account for business cycle regularities.
    \item Hornstein, Krusell, Violante (2011): can't account for wage dispersion.
    \end{enumerate}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{A Model of Sequential Search}
  \begin{itemize}
  \item The first model we'll look at is called the ``McCall Model'' (McCall, 1970).
  \item Basic idea:
    \begin{enumerate}
    \item Workers can be in one of two states: employed or unemployed, with value functions $V, U$.
    \item Receive job offers at exogenous rate $\alpha$, no information about meeting prior.
    \item Once employed, workers remain at current job until unexogenously separated (no OTJS) at rate $\delta$.
    \item Exogenous distribution of wages, $w\in [\underline{w},\bar{w}], w\sim F(.)$.
    \item Linear utility: $u(c) = b$ or $u(c) = w$.
    \end{enumerate}
  \item Optimal policy is a ``reservation strategy,'' i.e., a lower bound on the wages a worker will accept out of unemployment.
  \item Why is $w_{R} > \underline{w}$?
  \item What is the source of wage dispersion?
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Discrete Time Formulation}
  \begin{itemize}
  \item Each agent wants to maximize his discounted present value of consumption:
    \begin{align}
      \max\E \sum_{t=0}^{\infty} \beta^t c_t \\
    \end{align}
  \item Some simplifying assumptions: $\alpha = 1, \delta = 0$.
  \item Unemployed Bellman:
    \begin{align}
      U &= b + \beta E[\max\{V, U\}]\\
      U &= b + \beta \int_{\underline{w}}^{\bar{w}}\max\{V, U\}dF(w)
    \end{align}
  \item Employed Bellman:
    \begin{align}
      V(w) &= w + \beta V(w)\\
      V(w) &= \frac{w}{1 - \beta}
    \end{align}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Reservation Strategy}
  \begin{itemize}
  \item The reservation strategy is the lowest wage a worker will accept to leave unemployment.
  \item i.e., $V(w_{R}) = U$.
  \item Unemployed Bellman:
    \begin{align}
      \rightarrow V(w_{R}) &= U = \frac{w_{R}}{1 - \beta}\\
      \rightarrow \frac{w_{R}}{1 - \beta} &= b + \beta \int_{\underline{w}}^{\bar{w}}\max\{V, U\}dF(w)\\
      \rightarrow \frac{w_{R}}{1 - \beta} &= b + \beta \int_{\underline{w}}^{\bar{w}}\max\{\frac{w}{1 - \beta}, \frac{w_{R}}{1 - \beta}\}dF(w)\\
      \rightarrow (1 - \beta)w_{R} &= (1 - \beta)b + \beta \int_{\underline{w}}^{\bar{w}}\max\{w - w_{R}, 0\}dF(w)\\
      \rightarrow w_{R} &= b + \frac{\beta}{1 - \beta} \int_{\underline{w}}^{\bar{w}}\max\{w - w_{R}, 0\}dF(w)
    \end{align}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Reservation Strategy}
  \begin{itemize}
  \item Reservation strategy:
    \begin{align}
      w_{R} &= b + \frac{\beta}{1 - \beta} \int_{\underline{w}}^{\bar{w}}\max\{w - w_{R}, 0\}dF(w)
    \end{align}
  \item Integrate by parts:
\begin{align*}
\int u dv &= uv - \int v du. \\
\int\limits_{w_R}^{\overline{w}} (w-w_R)dF(w) \qquad &\Longrightarrow \qquad
\begin{tabular}{ll}
$u \: = w-w_R$ & $v \: = F(w)$ \\
$du = dw$ & $dv = dF(w)$
\end{tabular}\\
\int\limits_{w_R}^{\overline{w}} (w-w_R)dF(w) & = (w-w_R) F(w) \bigg|_{w_R}^{\overline{w}} - \int\limits_{w_R}^{\overline{w}} F(w)dw \\
& = \int\limits_{w_R}^{\overline{w}}[1-F(w)]dw
\end{align*}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Reservation Strategy}
  \begin{itemize}
  \item Reservation strategy:
    \begin{align}
      w_{R} &= b + \frac{\beta}{1 - \beta}\int\limits_{w_R}^{\overline{w}}[1-F(w)]dw
    \end{align}
  \item How would we solve for this?
  \item Assume a functional form for the distribution.
  \item Use root-finding algorithm to find $w_{R}$ st:
    \begin{align}
            w_{R} - b + \frac{\beta}{1 - \beta}\int\limits_{w_R}^{\overline{w}}[1-F(w)]dw  = 0
    \end{align}
  \item Sounds like a good homework assignment!
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Discrete Time Formulation}
  \begin{itemize}
  \item Search models typically written in continuous time.
  \item Easier to work with analytically.
  \item Discrete time Bellman equation for Unemployment:
    \begin{align}
      (1 + rdt)U &= bdt + \alpha dt E[\max\{V, U\}] + (1 - \alpha dt) U\\
      (r + \alpha)dtU &= bdt + \alpha dt E[\max\{V, U\}]\\
      U &= \frac{bdt + \alpha dt E[\max{V, U}]}{(r + \alpha)dt}
    \end{align}
  \item Taking limit as $dt\rightarrow 0$:
    \begin{align}
      \der{Num.}{dt} &= b + \alpha E[\max\{V, U\}]\\
      \der{Denom.}{dt} &= {(r + \alpha)}\\
      \Rightarrow U &= \frac{b + \alpha E[\max\{V, U\}]}{r + \alpha}
    \end{align}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Existence and Uniqueness}
  \begin{itemize}
  \item For simplicity, assume $V = \frac{w}{r}$, i.e. $\delta = 0$. Then,
    \begin{align}
      U &= \frac{b}{r + \alpha} + \frac{\alpha}{r + \alpha} E[\max\{\frac{w}{r}, U\}]
    \end{align}
  \item $U = T(U)$ is a contraction:
    \begin{enumerate}
    \item Discounting: $(\frac{\alpha}{r + \alpha} < 1)$.
    \item Monotonicity: $T(U)$ is nondecreasing in $U$.
    \end{enumerate}
  \item By Blackwell's Sufficient Conditions, this is a contraction with a unique fixed-point.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Continuous Time Formulation}
  \begin{itemize}
  \item Generally, we will use the continuous time Bellman in its ``asset value'' formulation:
    \begin{align}
      U &= \frac{b + \alpha E[\max\{V, U\}]}{r + \alpha}\\
      (r + \alpha)U &= b + \alpha E[\max\{V, U\}]\\
      rU &= b + \alpha E[\max\{V - U, 0\}]\\
      rU &= b + \alpha \int_{\underline{w}}^{\bar{w}}\max\{V - U, 0\}dF(w)
    \end{align}
  \item Employment:
    \begin{align}
      rV(w) &= w - \delta (V(w) - U)
    \end{align}
  \item Jobs lost at rate $\delta$.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Reservation Strategy}
  \begin{itemize}
  \item Reservation wage: $V(w_{R}) = U$:
    \begin{align}
      rV(w_{R}) &= w_{R} - \delta (V(w_{R}) - U)\\
      V(w_{R}) &= U = \frac{w_{R}}{r}\\
      \Rightarrow w_{R} &= b + \alpha \int_{\underline{w}}^{\bar{w}}\max\{V - U, 0\}dF(w)\\
       &= b + \alpha \int_{\underline{w}}^{\bar{w}}\max\{\frac{w + \delta U}{r + \delta} - \frac{w_{R}}{r}, 0\}dF(w)\\
       &= b + \alpha \int_{\underline{w}}^{\bar{w}}\max\{\frac{w + \delta \frac{w_{R}}{r}}{r + \delta} - \frac{w_{R}}{r}, 0\}dF(w)\\
      &= b + \frac{\alpha}{r + \delta} \int_{\underline{w}}^{\bar{w}}\max\{w - w_{R}, 0\}dF(w)
    \end{align}
  \item Note: if $\delta = 0$, identical to discrete time formulation.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Reservation Strategy II}
  \begin{itemize}
  \item Truncating and integrating by parts:
    \begin{align}
      w_{R} &= b + \frac{\alpha}{r + \delta}  \int_{\underline{w}}^{\bar{w}}\max\{w - w_{R}, 0\}dF(w)\\
      w_{R} &= b + \frac{\alpha}{r + \delta} \int_{w_{R}}^{\bar{w}}(w - w_{R})dF(w)
    \end{align}
    \begin{align}
      \int_{w_{R}}^{\bar{w}}(w - w_{R})dF(w) &= (w - w_{R})F(w)|_{w_{R}}^{\bar{w}} - \int_{w_{R}}^{\bar{w}}F(w)dw\\
                                             &= (\bar{w} - w_{R})F(\bar{w}) - \cancel{(w_{R} - w_{R})}F(w_{R}) \\&- \int_{w_{R}}^{\bar{w}}F(w)dw
    \end{align}
    \begin{align}
      \rightarrow       w_{R} &= b + \frac{\alpha}{r + \delta} \int_{w_{R}}^{\bar{w}}[1 - F(w)]dw
    \end{align}
  \end{itemize}
\end{frame}

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

\section{Dynamics and Predictions} % 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{Hazard Rate}
  \begin{itemize}
  \item What is the hazard rate of unemployment?
  \item Rate of leaving unemployment at time t.
  \begin{align}
    H_{u}(t) &= \alpha\int_{w_{R}}^{\bar{W}}dF(w)\\
         &= \alpha(F(\bar{w}) - F(w_{R}))\\
         &= \underbrace{\alpha}_{Meeting Rate} \underbrace{(1 - F(w_{R}))}_{Selectivity}
  \end{align}
\item Note, almost every search model generates a hazard composed of the product of a meeting probability and worker selectivity.
\item This is important to remember.
\item Hazard rate of employment (leaving employment for unemployment)?
  \begin{align}
    H_{e}(t) &= \delta
  \end{align}
\item Because separations are independent of state.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Dynamics of Unemployment}
  \begin{itemize}
  \item Use hazard rates to understand dynamics and steady-state.
  \item What does the model predict about employment and unemployment?
  \begin{align}
    \dot{u} &= \delta (1 - u) - \alpha (1 - F(w_{R}))u\\
    \dot{e} &= \alpha (1 - F(w_{R}))(1 - e) - \delta e
  \end{align}
\item Steady-state: $\dot{u} = 0$, $\dot{e} = 0$:
  \begin{align}
    0 &= \delta (1 - u) - \alpha (1 - F(w_{R}))u\\
    \rightarrow u &= \frac{\delta}{\delta + \alpha (1 - F(w_{R}))}\\
    0 &= \alpha (1 - F(w_{R}))(1 - e) - \delta e\\
   \rightarrow e &= \frac{\alpha (1 - F(w_{R}))}{\alpha (1 - F(w_{R})) + \delta}
  \end{align}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Wage Dispersion}
  \begin{itemize}
  \item What is wage dispersion in this model?
  \item Not {\it exactly} wage distribution.
  \item Workers reject some of the jobs posted.
  \item We will go through this next time.
  \end{itemize}
\end{frame}

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

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


\begin{frame}
  \frametitle{Next Time}
  \begin{itemize}
  \item Most likely DMP model.
  \item Between now and then:
    \begin{enumerate}
    \item Access the campus storage/cluster.
    \item Run some example code.
    \item Start working on your empirical regularities project.
    \end{enumerate}
  \end{itemize}
\end{frame}


\end{document}
