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%	TITLE PAGE
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\title[]{Quantitative Macro-Labor:\\ Wage Dispersion and Comparative Statics} % 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 2026} % 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 Present introduction/outline for research paper next Tuesday.
  \item This could be as little as a single slide (example on next slide).
  \item Introduction/outline due following week (2-3 pages).
  \item Will try to return comments to you today.
  \item I've been guaranteed that you have cluster access now.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Example Outline Project Slide}
  \begin{itemize}
  \item Research Idea/Topic: The Impact of AI on Human Capital.
  \item Question: ``AI use is difficult to detect and creates long-term learning deficiencies with undirected use. How will this affect primary, secondary, and higher education, and what will the impact on aggregate human capital be?''
  \item Data and Empirical Approach: Use IPEDS to estimate contemporaneous impact of AI on cognition. Use pandemic to infer longer-term effects.
  \item Theory: Build human capital model (Ben-Porath) with costly human capital acquisition. GE effects: expectations about future labor market?
  \item The written outline is just a longer version of this where I would expand the question, describe the data in detail as well as the empirical specification, and explicitly describe the trade-offs that I'd like the theory to capture.
  \item Just one or two slides.
  \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{Recap: The McCall Model}
  \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 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 $w_{R} > \underline{w}$?
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Model and Reservation Strategy}
  \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 Reservation strategy:
    \begin{align}
      w_{R} &= b + \frac{\alpha}{r + \delta} \int_{w_{R}}^{\bar{w}}[1 - F(w)]dw
    \end{align}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Reservation Strategy II}
  \begin{itemize}
  \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}{r + \delta} \int_{w_{R}}^{\bar{w}}(w - w_{R})dF(w)
    \end{align}
  \item Assume that the distribution of wage offers is uniform.
  \item What is the conditional expectation of a truncated uniform random variable? $E[w- w_{R}|w\geq w_{R}] = \frac{\bar{w} - w_{R}}{2}$
  \item What is the probability of drawing from the truncated part of the offer distribution? $P(w\geq w_{R}) = \frac{\bar{w} - w_{R}}{\bar{w} - \underline{w}}$.
    \begin{align}
      w_{R} &= b + \frac{\alpha}{r + \delta}\frac{\bar{w} - w_{R}}{2}\frac{\bar{w} - w_{R}}{\bar{w} - \underline{w}}\\
      w_{R} &= b + \frac{\alpha}{r + \delta}\frac{(\bar{w} - w_{R})^{2}}{2(\bar{w} - \underline{w})}
    \end{align}
  \item (yes, this is the same as if you integrate the option value)
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Reservation Strategy II}
  \begin{itemize}
  \item Reservation strategy:
    \begin{align}
      w_{R} &= b + \frac{\alpha}{r + \delta}\frac{(\bar{w} - w_{R})^{2}}{2(\bar{w} - \underline{w})}\\
      w_{R} &= b + \frac{\alpha}{r + \delta}\frac{\bar{w}^{2} - \bar{w}w_{R} + w_{R}^{2}}{2(\bar{w} - \underline{w})}\\
      0 &= b + \frac{\alpha}{r + \delta}\frac{\bar{w}^{2}}{2(\bar{w} - \underline{w})} - (1 + \frac{\alpha}{r + \delta}\frac{\bar{w}}{2(\bar{w} - \underline{w})})w_{R} \\ &+ \frac{\alpha}{r + \delta}\frac{1}{2(\bar{w} - \underline{w})}w_{R}^{2}
    \end{align}
  \item Apply quadratic formula and choose root st $w_{R}\in [0, 1]$
  \begin{align}\hspace{-10mm}
      \frac{(1 + \frac{\alpha}{r + \delta}\frac{\bar{w}}{2(\bar{w} - \underline{w})})\pm\sqrt{(1 + \frac{\alpha}{r + \delta}\frac{\bar{w}}{2(\bar{w} - \underline{w})})^{2} - 4(b + \frac{\alpha}{r + \delta}\frac{\bar{w}^{2}}{2(\bar{w} - \underline{w})})(\frac{\alpha}{r + \delta}\frac{1}{2(\bar{w} - \underline{w})})}}{2(b + \frac{\alpha}{r + \delta}\frac{\bar{w}^{2}}{2(\bar{w} - \underline{w})})}
    \end{align}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Reservation Strategy III}
  \begin{itemize}
  \item Reservation strategy:
  \begin{align}\hspace{-10mm}
      \frac{(1 + \frac{\alpha}{r + \delta}\frac{\bar{w}}{2(\bar{w} - \underline{w})})\pm\sqrt{(1 + \frac{\alpha}{r + \delta}\frac{\bar{w}}{2(\bar{w} - \underline{w})})^{2} - 4(b + \frac{\alpha}{r + \delta}\frac{\bar{w}^{2}}{2(\bar{w} - \underline{w})})(\frac{\alpha}{r + \delta}\frac{1}{2(\bar{w} - \underline{w})})}}{2(b + \frac{\alpha}{r + \delta}\frac{\bar{w}^{2}}{2(\bar{w} - \underline{w})})}
    \end{align}
  \item Let's just pick some values for the parameters (assume monthly calibration):
    \begin{enumerate}
    \item $w\sim U[0,1]$
    \item $\alpha = 0.43:$ avg. mon. U-E (this isn't right. Why?)
    \item $\delta = 0.03:$ avg. mon. E-U
    \item $r = 0.0041:$ ann. int. rate
    \item $b = 0.4:$ UI rep. rate
    \end{enumerate}
  \item $w_{R}\in\{1.31,0.72\}$
  \item I'm a little skeptical of these results, but you get the idea.
  \end{itemize}
\end{frame}

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

\section{Comparative Statics} % 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 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{What can we say about an increase in UI?}
  \begin{itemize}
  \item Whenever we write down a model, we have created a laboratory.
  \item Let's run experiments with it!
  \item What will happen to wages and unemployment if UI $b$ increases?
  \item For wages, all we need to know is the change in the reservation strategy:
    \begin{align}
      \der{w_{R}}{b} &= 1 + \frac{\alpha}{r + \delta} \der{\int_{w_{R}}^{\bar{w}}[1 - F(w)]dw}{w_{R}}\der{w_{R}}{b}
    \end{align}
  \item Leibniz's integral rule:
    \begin{align}
      \der{}{x}\int_{a(x)}^{b(x)}f(x,t)dt &= f(x,b(x))\der{b(x)}{x} - f(x,a(x))\der{a(x)}{x} \\&+ \int_{a(x)}^{b(x)}\der{}{x}f(x,t)dt
    \end{align}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{$\der{w_{R}}{b}$}
  \begin{itemize}
  \item For wages, all we need to know is the change in the reservation strategy:
    \begin{align}
      \der{w_{R}}{b} &= 1 + \frac{\alpha}{r + \delta} \der{\int_{w_{R}}^{\bar{w}}[1 - F(w)]dw}{w_{R}}\der{w_{R}}{b}
    \end{align}
  \item Leibniz's integral rule:
    \begin{align*}\tiny\hspace{-15mm}
      \der{}{x}\int_{a(x)}^{b(x)}f(x,t)dt &= f(x,b(x))\der{b(x)}{x} - f(x,a(x))\der{a(x)}{x} + \int_{a(x)}^{b(x)}\der{f(x,t)}{x}dt
    \end{align*}\hspace{-10mm}
    \begin{align}\tiny
      \der{w_{R}}{b} &= 1 + \frac{\alpha}{r + \delta}(\cancel{[1 - F(\bar{w})]}\cancel{\der{\bar{w}}{w_{R}}} \nonumber\\&- [1 - F(w_{R})]\der{w_{R}}{w_{R}} + \cancel{\int_{w_{R}}^{\bar{w}}\der{[1 - F(w)]}{w_{R}}})\nonumber\\
      \der{w_{R}}{b} &= 1 - \frac{\alpha}{r + \delta}[1 - F(w_{R})]\der{w_{R}}{b}\nonumber\\
      \der{w_{R}}{b} &= \frac{r + \delta}{r + \delta + \alpha[1 - F(w_{R})]} < 1
    \end{align}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{What about unemployment?}
  \begin{itemize}
  \item Now, $\der{u}{b}$.
  \item Let's find the semi-elasticity: $\der{log(u)}{b}$
  \begin{align}
    ln(u) &= ln(\delta) - ln(\delta + \alpha (1 - F(w_{R})))\\
    \der{ln(u)}{b} &= \frac{\alpha f(w_{R})\der{w_{R}}{b}}{\delta + \alpha (1 - F(w_{R}))}
  \end{align}
\item Unemployment clearly increases.
\item More interesting: separation rate ($\delta$) and offer arrival rate ($\alpha$)
\item Why? Predictions are unclear.
\item If $\alpha\uparrow$, find jobs faster, but also sample better jobs more often.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{$\der{w_{R}}{\alpha}$}
  \begin{itemize}
  \item For wages, all we need to know is the change in the reservation strategy:
    \begin{align}
      \der{w_{R}}{\alpha} &= \underbrace{\frac{1}{r + \delta}\int_{w_{R}}^{\bar{w}}[1 - F(w)]dw}_{Match\;Rate} - \underbrace{\frac{\alpha}{r + \delta}[1 - F(w_{r})]\der{w_{R}}{\alpha}}_{Selectivity}\\
      \der{w_{R}}{\alpha} &= \frac{\int_{w_{R}}^{\bar{w}}[1 - F(w)]dw}{r + \delta + \alpha[1 - F(w_{r})]}
    \end{align}
  \item Now the semi-elasticity: $\der{log(u)}{\alpha}$
  \begin{align}
    ln(u) &= ln(\delta) - ln(\delta + \alpha (1 - F(w_{R})))\\
    \der{ln(u)}{\alpha} &= \underbrace{\frac{\alpha f(w_{R})\der{w_{R}}{\alpha}}{\delta + \alpha (1 - F(w_{R}))}}_{Selectivity} - \underbrace{\frac{(1 - F(w_{R}))}{\delta + \alpha (1 - F(w_{R}))}}_{Match\;Rate}
  \end{align}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Log-Concavity}
  \begin{align}
    \der{ln(u)}{\alpha} &= \frac{\alpha f(w_{R})\frac{\int_{w_{R}}^{\bar{w}}[1 - F(w)]dw}{r + \delta + \alpha[1 - F(w_{r})]}}{\delta + \alpha (1 - F(w_{R}))} - \frac{(1 - F(w_{R}))}{\delta + \alpha (1 - F(w_{R}))}
  \end{align}
  \begin{itemize}
  \item Uh oh... how are we going to sign this?
  \item Properties of log-concave distributions (where $F(x)$ is log-concave):
    \begin{enumerate}
    \item $F(x)$ log-concave $\rightarrow\int F(x)$ log-concave.
    \item $F(x)$ log-concave $\rightarrow\der{F}{x}$ log-concave.
    \item $F(x)F''(x)\leq (F'(x))^{2}$
    \end{enumerate}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Log-Concavity}
  \begin{itemize}
  \item Properties of log-concave distributions (where $F(x)$ is log-concave):
    \begin{enumerate}
    \item $F(x)$ log-concave $\rightarrow\int F(x)$ log-concave.
    \item $F(x)$ log-concave $\rightarrow\der{F}{x}$ log-concave.
    \item $F(x)F''(x)\leq (F'(x))^{2}$
    \end{enumerate}
  \item $(\delta + \alpha (1 - F(w_{R}))) > 0$
  \begin{align}
    \der{ln(u)}{\alpha} &\propto \alpha f(w_{R})\frac{\int_{w_{R}}^{\bar{w}}[1 - F(w)]dw}{r + \delta + \alpha[1 - F(w_{r})]} - (1 - F(w_{R}))\\
                        &\propto \alpha f(w_{R})\int_{w_{R}}^{\bar{w}}[1 - F(w)]dw - (1 - F(w_{R}))^{2} < 0
  \end{align}
\item By the third property of log-concave distributions.
  \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{``Estimation''/Calibration}
  \begin{itemize}
  \item Earlier, I picked some parameters from Hornstein, Krusell, and Violante (``Calibrated Example'')
  \item If we want to match this model to the data, what targets can we use?
  \item Unconditional moments (i.e., population averages):
    \begin{itemize}
    \item Hazard rates (U-E, U-E)
    \item Employment rates (e, u)
    \item Wage distribution
    \end{itemize}
  \item How many moments do we need?
    \begin{itemize}
    \item $\delta$: separation rate
    \item $\alpha$: match rate
    \item $F(w)$: distribution function
    \end{itemize}
  \item Can we separately (ex-ante) identify them?
  \item Particularly, what can we use to identify $\alpha$ and $F(w)$?
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{``Estimation''/Calibration II}
  \begin{itemize}
  \item What targets can we use to discipline model?
  \item Unconditional moments (i.e., population averages):
    \begin{itemize}
    \item Hazard rates (U2E, E2U)
    \item Employment rates (e, u)
    \item Wage distribution
    \end{itemize}
  \item What should we match?
  \item Depends on what we are after:
    \begin{itemize}
    \item Transition rates: don't target the transition rates.
    \item Wage distribution: don't target the wage distribution.
    \end{itemize}
  \item What time period should we use?
  \item Steady-state: time-independent .
  \item We could pick any time interval and get same steady-state.
  \item But, pick monthly.
  \item Return to calibration momentarily.
  \end{itemize}
\end{frame}

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

\section{Wage Dispersion} % 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 What does the McCall model say is the source of wage dispersion?
  \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 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{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}}
    \end{align}
  \begin{itemize}
  \item It's hard to separately identify the {\it offer} distribution and the {\it accepted} offer or wage distribution.
  \item This expression ignores the distinction: $\alpha(1 - F(w_{R})) = H_{u}$.
  \item We just need the observed hazard, and can plug in for values.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Calibration II}
  \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{Conclusion}
% ------------------------------------------------


\begin{frame}
  \frametitle{Next Time}
  \begin{itemize}
  \item Extensions of the McCall model: On-the-Job Search.
  \item Between now and then:
    \begin{enumerate}
    \item Access the campus storage/cluster.
    \item Run some example code.
    \item Start research proposal and one slide presentation.
    \end{enumerate}
  \end{itemize}
\end{frame}


\end{document}