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% ----------------------------------------------------------------------------------------
%	TITLE PAGE
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\title[]{Quantitative Macro-Labor:\\ Extending the McCall Model with On-the-Job 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 2022} % 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 Chance I will have jury duty next week. Possibly means no class all next week.
  \item Today: On-the-Job Search (Burdett-Mortensen, 1998)
  \item You should have logged into the cluster/storage by now.
  \item Set up your own folder under ``student\_folders''. Use the format first initial-last name (i.e., bgriffy).
  \item Never put spaces in your directories (use underscores \_)
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Running Code on the Cluster I}
  \begin{itemize}
  \item There is code to generate the following datasets:
    \begin{enumerate}
    \item PSID: you'll need Python and Stata, and run ``PSIDMaster.py'' after choosing variables.
    \item SIPP: you'll need Stata, run ``SIPPMaster.do''
    \item NLSY: pick variables from NLSY website.
    \end{enumerate}
  \item Run ``Bewley1986'' Matlab code as a test on cluster
  \item Please install the anaconda distribution of Python.
  \item Get Matlab from the UAlbany software system.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Running Code on the Cluster II}
  \begin{itemize}
  \item Open terminal where your ``slurm-file'' is located.
  \item Type ``sbatch slurmFileName'' and it will run.
  \item Change the email in the slurm file so that I don't get a bunch of emails.
  \item Run ``Bewley1986'' Matlab code as a test.
  \end{itemize}
\end{frame}

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

\section{Wage Dispersion in 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{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}
  \item McCall model: after we control for ability (or whatever), worker selectivity generates all wage dispersion.
  \item The more selective workers are, the smaller the dispersion is.
  \item Hornstein, Krusell, Violante: Simple models can't match dispersion.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{McCall Model Review}
  \begin{itemize}
  \item Model equilibrium characterized by two flow equations and a policy function: unemployment, employment and reservation wage.
  \item Unemployment:
    \begin{align}
      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 Reservation strategy:
    \begin{align}
      w_{R} &= b + \frac{\alpha}{r + \delta} \int_{w_{R}}^{\bar{w}}[1 - F(w)]dw\\
      w_{R} &= b + \frac{\alpha(1 - F(w_{R}))}{r + \delta} \int_{w_{R}}^{\bar{w}}(w - w_{R})\frac{dF(w)}{1 - F(w_{R})}
    \end{align}
  \end{itemize}
\end{frame}

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

% \begin{frame}
%   \frametitle{A Notion of Wage Dispersion}
%   \begin{itemize}
%   \item Extremely clever paper: Hornstein, Krusell, Violante (2011).
%   \item Basic idea: use the mean-min (Mm) ratio for wage dispersion.
%   \item Almost every search model has expression for the Mm ratio.
%   \item Compare model Mm with data Mm.
%   \item Reservation strategy:
%     \begin{align}
%       w_{R} &= b + \frac{\alpha}{r + \delta} \int_{w_{R}}^{\bar{w}}(w - w_{R})dF(w)\\
%       w_{R} &= b + \frac{\alpha(1 - F(w_{R}))}{r + \delta} \int_{w_{R}}^{\bar{w}}(w - w_{R})\frac{dF(w)}{1 - F(w_{R})}\\
%       w_{R} &= b + \frac{\alpha(1 - F(w_{R}))}{r + \delta} \int_{w_{R}}^{\bar{w}}(w - w_{R})\frac{dF(w)}{1 - F(w_{R})}
%     \end{align}
%   \item Exp. of a truncated random variable: $E[w|w\geq w_{R}] = \hat{w}$
%     \begin{align}
%       \rightarrow w_{R} &= b + \frac{\alpha(1 - F(w_{R}))}{r + \delta}[\hat{w} - w_{R}]
%     \end{align}
%   \end{itemize}
% \end{frame}

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

% \begin{frame}
%   \frametitle{The Mean-Min Ratio}
%   \begin{itemize}
%   \item Exp. of a truncated random variable: $E[w|w\geq w_{R}] = \hat{w}$
%     \begin{align}
%       \rightarrow w_{R} &= b + \frac{\alpha(1 - F(w_{R}))}{r + \delta}[\hat{w} - w_{R}]
%     \end{align}
%   \item Average UI in the economy: $b = \rho\hat{w}$
%   \item Reservation strategy:
%     \begin{align}
%       w_{R} &= \rho\hat{w} + \frac{\alpha(1 - F(w_{R}))}{r + \delta}[\hat{w} - w_{R}]
%     \end{align}
%   \item What is minimum wage in this economy? $w_{R}$ of course!
%     \begin{align}
%       (\rho + \frac{\alpha(1 - F(w_{R}))}{r + \delta})\hat{w} &= (1 + \frac{\alpha(1 - F(w_{R}))}{r + \delta})w_{R}\\
%       \rightarrow \frac{\hat{w}}{w_{R}} &= \frac{1 + \frac{\alpha(1 - F(w_{R}))}{r + \delta}}{\rho + \frac{\alpha(1 - F(w_{R}))}{r + \delta}}
%     \end{align}
%   \item Good news: $\rho < 1\rightarrow$ mean wage is greater than $w_{R}$.
%   % \item What is {\it incredibly} useful (empirically) about this formulation?
%   \end{itemize}
% \end{frame}

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

% \begin{frame}
%   \frametitle{Calibration}
%   \begin{itemize}
%   \item Parameters can be calibrated directly from observed data (or observed from HKV):
%     \begin{enumerate}
%     \item $\alpha(1 - F(w_{R})) = 0.43$: avg. mon. U-E (HKV)
%     \item $\delta = 0.03$: avg. mon. E-U (HKV)
%     \item $r = 0.0041$: ann. int. rate (HKV)
%     \item $\rho = 0.4$: UI rep. rate (HKV)
%     \end{enumerate}
%     \begin{align}
%       \frac{\hat{w}}{w_{R}} &= \frac{1 + \frac{\alpha(1 - F(w_{R}))}{r + \delta}}{\rho + \frac{\alpha(1 - F(w_{R}))}{r + \delta}} = \frac{1 + \frac{0.43}{0.0341}}{0.4 + \frac{0.43}{0.0341}} = 1.046
%     \end{align}
%   \item Great! What does this tell us?
%   \item The McCall model predicts Mm wage dispersion of 4.6\%.
%   \end{itemize}
% \end{frame}

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

% \begin{frame}
%   \frametitle{Wage Dispersion}
%   \begin{itemize}
%   \item The McCall model predicts Mm wage dispersion of 4.6\%.
% \begin{center}\includegraphics[width=0.6\textwidth]{WageHistogram.png}\end{center}
% \item HKV: Mm ratio is roughly 2.
% \item What does this mean?
%   \end{itemize}
% \end{frame}

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

\section{The Burdett-Mortensen (1998) Model}
% ------------------------------------------------

\begin{frame}
  \frametitle{The Burdett-Mortensen Model}
  \begin{itemize}
  \item What is an (one of many) important and realistic feature of the labor market missing in the standard McCall model?
  \item The ability to search while employed.
  \item Some statistics:
    \begin{enumerate}
    \item 50\% of all hires are job-to-job hires (Census)
    \item Movement up job ladder accounts for 50\% of wage growth for young workers (Topel and Ward, 1992)
    \item 70\% of fall in hires during Great Recession was J2J.
    \end{enumerate}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Job-to-Job Flows}
  \begin{itemize}
  \item J2J big fraction of job flows (Hahn et al, 2017)
  \end{itemize}
\begin{center}\includegraphics[width=0.6\textwidth]{Hires_Flows.png}\end{center}
\end{frame}

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

\begin{frame}
  \frametitle{Frictional Earnings Growth}
  \begin{itemize}
  \item And earnings growth (Hahn et al, 2017)
  \end{itemize}
\begin{center}\includegraphics[width=0.6\textwidth]{EarningsGrowthSources.png}\end{center}
\end{frame}

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

\begin{frame}
  \frametitle{The Burdett-Mortensen OTJS Model}
  \begin{itemize}
  \item Extend McCall model to include ability to search while employed.
  \item (We'll stick with our notation rather than using theirs)
  \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 wages, i.e., a given distribution of wages, $w\in [\underline{w},\bar{w}], w\sim F(.)$.
    \item Unemployed receive job offers at exogenous rate $\alpha$, no prior info.
    \item Employed receive job offers at exogenous rate $\lambda$, no prior info.
    \item Separate two ways: exogenously (rate $\delta$) and via thru OTJS (rate $\lambda [1 - F(w)]$)
    \item Linear utility: $u(c) = b$ or $u(c) = w$.
    \end{enumerate}
  \item Important assumption: outside offers are unverifiable.
  \item i.e., firms cannot respond to outside offers.
  \item Intuitively, are there parameter values for which $w_{R} = \underline{w}$?
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Ex Ante Homogeneous Model (``The BM Model'')}
  \begin{itemize}
  \item We'll see two key components to matching flows:
    \begin{enumerate}
    \item The job-finding rate $\alpha[1-F(w_{R})]$
    \item The worker-finding rate $l(w|w_{R},F)$
    \end{enumerate}
  \item Flow value of unemployment:
    \begin{align*}
      rU &= b + \alpha \int_{\underline{w}}^{\bar{w}}\max\{V(x) - U, 0\}dF(x)
    \end{align*}
  \item Employment:
    \begin{align*}
      rV(w) &= w + \lambda \int_{\underline{w}}^{\bar{w}}\max\{V(x) - V(w), 0\}dF(x) + \delta (U - V(w))
    \end{align*}
  \item Over what range will the employed individual accept a job?
  \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} + \lambda \int_{w_{R}}^{\bar{w}}(V(x) - V(w_{R}))dF(x) \nonumber\\&+ \delta (U - V(w_{R}))\\
      rV(w_{R}) &= rU = w_{R} + \lambda \int_{w_{R}}^{\bar{w}}(V(x) - V(w_{R}))dF(x)\\
      \Rightarrow rU - rV(w_{R}) &= 0 = b - w_{R} \nonumber\\&+ (\alpha - \lambda)\int_{w_{R}}^{\bar{w}}(V(x) - V(w_{R}))dF(x)\\
      \Rightarrow w_{R} &= b + (\alpha - \lambda)\int_{w_{R}}^{\bar{w}}(V(x) - V(w_{R}))dF(x)
      %  &= 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 If $\alpha = \lambda$, what is the interpretation?
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Reservation Strategy II}
  \begin{itemize}
  \item Integration by parts:
    \begin{align*}
      w_{R} &= b + (\alpha - \lambda)\int_{w_{R}}^{\bar{w}}(V(x) - V(w_{R}))dF(x)
    \end{align*}
    \begin{align*}
      \int_{w_{R}}^{\bar{w}}(V(x) &- V(w_{R}))dF(x) = (V(x) - V(w_{R}))F(x)|_{w_{R}}^{\bar{w}} \\&- \int_{w_{R}}^{\bar{w}}V'(x)F(x)dx\\
                                             &= (V(\bar{w}) - V(w_{R}))F(\bar{w}) - \int_{w_{R}}^{\bar{w}}V'(x)F(x)dx\\
                                             &= \int_{w_{R}}^{\bar{w}}V'(x)[1 - F(x)]dx
    \end{align*}
    \begin{align*}
      \rightarrow       w_{R} &= b + (\alpha - \lambda) \int_{w_{R}}^{\bar{w}}V'(x)[1 - F(x)]dx
    \end{align*}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Reservation Strategy III}
  \begin{itemize}
    \begin{align*}
      w_{R} &= b + (\alpha - \lambda) \int_{w_{R}}^{\bar{w}}V'(x)[1 - F(x)]dx
    \end{align*}
  \item What is the derivative of the employed value function?
    \begin{align*}
      rV(w) &= w + \lambda \int_{\underline{w}}^{\bar{w}}\max\{V(x) - V(w), 0\}dF(x) + \delta (U - V(w))\\
      \rightarrow r\der{V}{w} &= dw - \lambda[1 - F(w_{R})]\der{V}{w} - \delta\der{V}{w}\\
      \rightarrow \der{V}{w} &= \frac{dw}{r + \delta + \lambda[1 - F(w_{R})]}
    \end{align*}
  \item Thus, the reservation wage is
    \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{Steady-States}
  \begin{itemize}
  \item Flow unemployment equation:
    \begin{align*}
      \dot{u} &= \delta(1 - u) - \alpha[1 - F(w_{R})]u\\
      u &= \frac{\delta}{\delta + \alpha[1 - F(w_{R})]}
    \end{align*}
  \item Wage distribution?
  \item Key feature of this model: can characterize the steady-state wage distribution from model primitives.
  \item Define the CDF of employed wages at time t, $G(w, t)$ and population wage distribution $G(w, t)(1 - u(t))$:
    \begin{align*}
      \der{G(w,t)(1 - u(t))}{t}&=\alpha\max\{F(w) - F(w_{R}),0\}u(t) \nonumber\\&- [\delta + \lambda(1 - F(w))]G(w,t)(1 - u(t))
    \end{align*}
  \item $G(w,t)$: prop. of workers receiving no greater than $w$.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Wage Dispersion in Steady-State}
  \begin{itemize}
  \item In steady-state:
    \begin{align*}
      \der{G(w,t)(1 - u(t))}{t}&=0=\alpha\max\{F(w) - F(w_{R}),0\}u(t) \nonumber\\&- [\delta + \lambda(1 - F(w))]G(w,t)(1 - u(t))
    \end{align*}
  \item For $w\geq w_{R}$:
    \begin{align*}
      [\delta + \lambda(1 - F(w))]G(w)(1 - u(t))&=\alpha[F(w) - F(w_{R})]u(t)
    \end{align*}
    \begin{align*}
      \rightarrow G(w) &=\frac{\alpha(F(w) - F(w_{R}))}{[\delta + \lambda(1 - F(w))]}\frac{u(t)}{1 - u(t)}
    \end{align*}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Wage Dispersion in Steady-State}
  \begin{itemize}
  \item Combining unemployment and the wage distribution:
    \begin{align*}
      G(w) &=\frac{\alpha(F(w) - F(w_{R}))}{[\delta + \lambda(1 - F(w))]}\frac{u(t)}{1 - u(t)}\\
      u &= \frac{\delta}{\delta + \alpha[1 - F(w_{R})]}\\
      \rightarrow G(w) &=\frac{\alpha(F(w) - F(w_{R}))}{[\delta + \lambda(1 - F(w))]}\frac{\delta}{\alpha[1 - F(w_{R})]}\\
             &=\frac{\delta(F(w) - F(w_{R}))}{[\delta + \lambda(1 - F(w))][1 - F(w_{R})]}
    \end{align*}
  \item Intuitively, what does this say?
  \item What does $\lambda = 0$ imply about this distribution?
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{The Firm}
  \begin{itemize}
  \item What role does the firm play in all of this?
  \item How will it choose which wages to post?
  \item Firm profit function:
    \begin{align*}
      \pi &= \max_{w} (p - w)l(w|w_{R}, F)\\
      \pi(w|w_{R},F) &= (p - w)l(w|w_{R}, F)
    \end{align*}
  \item What is $l$? The probability that a firm will meet a worker if it posts a wage $w$, given
    \begin{enumerate}
    \item the reservation wage
    \item the distribution of other firm's wages
    \end{enumerate}
  \item This is basically a free entry condition.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Worker Finding Rate}
  \begin{itemize}
  \item What is $l(w|w_{R}, F)$?
  \item In steady-state, workers per firm earning a wage w, i.e., probability of finding a worker if you post a wage, $w$.
  \item Inflows:
    \begin{align*}
      \alpha u + (1 - u)\lambda G(w,t)
    \end{align*}
  \item Outflows:
    \begin{align*}
      (\delta + \lambda [1 - F(w)])
    \end{align*}
  \item Define $m = \frac{workers}{firms}$. Then,
    \begin{align*}
      (\delta + \lambda [1 - F(w)])l(w|w_{R}, F) &= m[\alpha u + (1 - u)\lambda G(w,t)]\\
      (\delta + \lambda [1 - F(w)])l(w|w_{R}, F) &= m\alpha \frac{\delta}{\delta + \alpha[1 - F(w_{R})]} \nonumber\\&+ \frac{m\alpha[1 - F(w_{R})]}{\delta + \alpha[1 - F(w_{R})]}\lambda G(w,t)]
    \end{align*}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Worker Finding Rate II}
  \begin{itemize}
  \item What is $l(w|w_{R}, F)$?
    \begin{align*}
      (\delta &+ \lambda [1 - F(w)])l(w|w_{R}, F) = m\alpha \frac{\delta}{\delta + \alpha[1 - F(w_{R})]} \nonumber\\&+ \frac{m\alpha[1 - F(w_{R})]}{\delta + \alpha[1 - F(w_{R})]}\lambda G(w)]\\
      l(w|w_{R}, F) &= m\alpha \frac{\delta + [1 - F(w_{R})]\lambda G(w)}{(\delta + \lambda [1 - F(w)])(\delta + \alpha[1 - F(w_{R})])}\\
      l(w|w_{R}, F) &= m\alpha \frac{\delta(1 + \lambda\frac{[F(w) - F(w_{R})]}{[\delta + \lambda(1 - F(w))]})}{(\delta + \lambda [1 - F(w)])(\delta + \alpha[1 - F(w_{R})])}\\
      l(w|w_{R}, F) &= m\alpha\delta \frac{\frac{\delta + \lambda(1 - F(w_{R}))}{\delta + \alpha(1 - F(w_{R}))}}{(\delta + \lambda[1 - F(w)])^{2}}
    \end{align*}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Equilibrium}
  \begin{itemize}
  \item From the paper:\\
{\it An equilibrium solution to the search and wage-posting game} outlined above can be described by a triple $(w_{R}, F, \pi)$ such that $w_{R}$, the common reservation wage of unemployed workers is the solution to the optimal stopping-time problem out of unemployment, and $\pi$ is the value function given by the optimal choice of $w$, and the distribution $F$ is such that
\begin{align*}
  (p - w)l(w|w_{R},F) &= \pi\hspace{2mm}\text{ for all w on the support of F}\\
  (p - w)l(w|w_{R},F) &\leq \pi\hspace{2mm}\text{ otherwise}
\end{align*}
\item This is essentially a ``free entry'' condition.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Analytical Characterizations}
  \begin{itemize}
  \item Assume that $\lambda = \alpha$, i.e., OTJS as efficient as off-the-job.
  \item Then, $w_{R} = b$.
    \begin{align*}
      l(w|w_{R}, F) &= m\alpha\delta \frac{\frac{\delta + \lambda(1 - F(w_{R}))}{\delta + \alpha(1 - F(w_{R}))}}{(\delta + \lambda[1 - F(w)])^{2}}\\
      l(w|w_{R}, F) &=  \frac{m\alpha\delta}{(\delta + \alpha[1 - F(w)])^{2}}\\
      \rightarrow (p - w)l(w|w_{R},F) & = (p - w)\frac{m\alpha\delta}{(\delta + \alpha[1 - F(w)])^{2}}
    \end{align*}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Analytical Characterizations}
  \begin{itemize}
  \item In equilibrium, profits are all equal: $(p - w)l(w|w_{R},F) & = (p - w_{R})l(w_{R}|w_{R},F)$
    \begin{align*}
            (p - w)l(w|w_{R},F) & = (p - w_{R})\frac{m\alpha\delta}{(\delta + \alpha[1 - F(w)])^{2}}
    \end{align*}\vspace{-5mm}
    \begin{align*}\hspace{-10mm}
      \rightarrow (p - w)\frac{m\alpha\delta}{(\delta + \alpha[1 - F(w)])^{2}} &= (p - w_{R})\frac{m\alpha\delta}{(\delta + \alpha[1 - \cancel{F(w_{R})}])^{2}}\\
      \rightarrow (\delta + \alpha[1 - F(w)])^{2} &= \frac{(p - w)}{p - w_{R}}(\delta + \alpha)^{2}\\
      \rightarrow [1 - F(w)] &= \frac{(\delta + \alpha)}{\alpha}[\frac{(p - w)}{p - w_{R}}]^{\frac{1}{2}} - \frac{\delta}{\alpha}\\
      \rightarrow F(w) &= 1 + \frac{\delta}{\alpha} - \frac{(\delta + \alpha)}{\alpha}[\frac{(p - w)}{p - w_{R}}]^{\frac{1}{2}}\\
      \rightarrow F(w) &= \frac{(\delta + \alpha)}{\alpha}(1 - [\frac{(p - w)}{p - w_{R}}]^{\frac{1}{2}})
    \end{align*}
  \item We've solved for the offered wage distribution!
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Equilibrium Wage Dispersion}
  \begin{itemize}
  \item Found that $G(w)$ given by
    \begin{align*}
      G(w) &= \frac{\delta(F(w) - F(w_{R}))}{[\delta + \alpha(1 - F(w))][1 - F(w_{R})]}\\
      G(w) &= \frac{\delta F(w)}{[\delta + \alpha(1 - F(w))]}
    \end{align*}\vspace{-5mm}
  \item And the offer distribution given by
    \begin{align*}
      F(w) &= \frac{(\delta + \alpha)}{\alpha}(1 - [\frac{(p - w)}{p - w_{R}}]^{\frac{1}{2}})
    \end{align*}
  \item Thus, the equilibrium wage distribution is given by
    \begin{align*}
      G(w) &= \frac{\delta \frac{(\delta + \alpha)}{\alpha}(1 - [\frac{(p - w)}{p - w_{R}}]^{\frac{1}{2}})}{[\delta + \alpha(1 - \frac{(\delta + \alpha)}{\alpha}(1 - [\frac{(p - w)}{p - w_{R}}]^{\frac{1}{2}}))]}
    \end{align*}
  \end{itemize}
\end{frame}

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

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


\begin{frame}
  \frametitle{Next Time}
  \begin{itemize}
  \item Disciplining wage contracts: risk aversion.
  \item Between now and then:
    \begin{enumerate}
    \item Access the campus storage/cluster.
    \item Run some example code.
    \item Create a folder in ``user\_folders'' called first initial-last name (i.e., bgriffy).
    \end{enumerate}
  \item Homework posted online. Due next Monday.
  \item Start your introduction/research proposal project.
  \item Will have presentations of these in a couple of weeks.
  \item I may not know until early next week whether or not I have jury duty.
  \end{itemize}
\end{frame}


\end{document}