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

\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 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 Briefly review Mortensen and Pissarides.
  \item Show extension incorporating endogenous separations.
  \end{itemize}
\end{frame}

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

% \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}
%   \end{itemize}
% \end{frame}

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


\title[]{Part-Time Employment and Labor Market Volatility} % The short title appears at the bottom of every slide, the full
% title is only on the title page

\author[Professor Griffy]{Pedro Gomis-Porqueras \inst{1} \and Ben Griffy \inst{2} \and Stan Rabinovich \inst{3}} % Your name
\institute[UAlbany] % Your institution as it will appear on the bottom of
% every slide, may be shorthand to save space
{
\inst{1} Deakin University \and
\inst{2} University at Albany, SUNY \and % Your institution for the title page
\inst{3} University of North Carolina - Chapel Hill \ % Your institution for the title page
}
\date{A long time ago} % Date, can be changed to a custom date

\begin{frame}
  \titlepage % Print the title page as the first slide
\end{frame}

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% ----------------------------------------------------------------------------------------
%	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{Overview}
  \begin{itemize}
  \item Here, we extend the Mortensen-Pissarides model to include heterogeneous match productivity.
  \item There will be an ``endogenous separation'' threshold and an ``endogenous promotion'' threshold.
  \item We generate the separate threshold through different costs.
  \item Ultimate goal of project is to show that acylical cost generates more procyclical employment and countercyclical unemployment.
  \item But here focus is on showing use of surplus and ex-post match heterogeneity to generate cool model features.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Paper Outline}
  \begin{itemize}
  \item Start with ``empirical regularities''
    \begin{itemize}
    \item What we will explore;
    \item Motivate our model construction.
    \end{itemize}
  \item Show existence of part-time jobs in steady-state model.
  \item Characterize productivity thresholds.
  \item Use discrete time version to simulate out of steady-state.
  \item (I will present this like a seminar to show a template for talk.)
  \item Bullet points vs. sentences:
    \begin{itemize}
    \item Sentence: contains a subject, verb, and a complete idea.
    \item Bullet point in talk: fits on one line.
    \end{itemize}
  \item The current version of the paper is much less theory, much more quantitative. This is more elegant.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Part and Full-Time Employment}\label{wealthandborrowingconstraints}
  \begin{itemize}
  \item Part-time employment large component of labor market:
    \begin{itemize}
    \item Part-time employment rate: 22\% (Prime age, 17\%)
    \item Ages 18-24: 35\%, up from 25\% a decade ago.
    \item Ages 55-64: 47\%
    \item Production: 40 hours/week for full-time, 28 hours/week for part-time
    \end{itemize}
  \item Search models rarely feature part-time employment.
\end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Cyclicality}\label{wealthandborrowingconstraints}
  \begin{itemize}
  \item Different cyclicality of part and full-time jobs.
    \begin{figure}[ht]
        \begin{minipage}[b]{0.465\linewidth}
            \centering
            \includegraphics[width=0.85\linewidth]{./EmpSQUARE.eps}
            \caption{Employment}
            \label{fig:a}
        \end{minipage}
        \hspace{0.5cm}
        \begin{minipage}[b]{0.45\linewidth}
            \centering
            \includegraphics[width=0.85\linewidth]{./FTEmpPTEmpUnempSQUARE.eps}
            \caption{Unemployment}
            \label{fig:b}
        \end{minipage}
    \end{figure}
\item Would PT improve fit?
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Cyclicality of Flows}\label{wealthandborrowingconstraints}
  \begin{itemize}
  \item Different cyclicality of part and full-time jobs.
    \begin{figure}[ht]
        \begin{minipage}[b]{0.465\linewidth}
            \centering
            \includegraphics[width=0.85\linewidth]{./FTPTFTUSQUARE.eps}
            \caption{Employment}
            \label{fig:a}
        \end{minipage}
        \hspace{0.5cm}
        \begin{minipage}[b]{0.45\linewidth}
            \centering
            \includegraphics[width=0.85\linewidth]{./PTFTPTUSQUARE.eps}
            \caption{Unemployment}
            \label{fig:b}
        \end{minipage}
    \end{figure}
\item Would PT improve fit?
  \end{itemize}
\end{frame}

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

% \begin{frame}
%   \frametitle{Unemployment Rates by Hours Worked}\label{lcprofilesbywealth}
% \begin{figure}
% \includegraphics[width=\linewidth]{UnempRatebyType.eps}
% \end{figure}
% \end{frame}

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

% \begin{frame}
%   \frametitle{Employment by Hours Worked}\label{lcprofilesbywealth}
% \begin{figure}
% \includegraphics[width=\linewidth]{EmpRatebyType.eps}
% \end{figure}
% \end{frame}

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

% \begin{frame}
%   \frametitle{Wages}\label{wealthandborrowingconstraints}
%   \begin{itemize}
%   \item Reported benefits appear acyclical
% \begin{figure}
% \includegraphics[width=\linewidth]{WagePrimeAge.eps}
% \end{figure}
% \end{itemize}
% \end{frame}

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

% \begin{frame}
%   \frametitle{Costs}\label{wealthandborrowingconstraints}
%   \begin{itemize}
%   \item Employers face costs outside of wages for workers.
%   \item Acyclical costs can affect predicted volatility in search and matching models (Pissarides, 2009).
%   \item Full-time and part-time employment involve different costs and contract obligations:
%     \begin{itemize}
%     \item paid time off: $\approx 12$ days/year for FT (90\%); $\approx 3$ days/year for PT (35\%).
%     \item access to healthcare \& other benefits: 85\% of FT; 19\% of PT.
%     \item access to employee retirement programs: 81\% of FT; 38\% of PT.
%     \end{itemize}
% \end{itemize}
% \end{frame}

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

% \begin{frame}
%   \frametitle{Employer Reported Benefit Costs}\label{wealthandborrowingconstraints}
%   \begin{itemize}
%   \item Data not great (annual from 1994-2001, quarterly after).
%   \item Does not appear to be cyclical.
% \begin{figure}
% \includegraphics[width=\linewidth]{Employment_TotalBenefits.eps}
% \end{figure}
% \end{itemize}
% \end{frame}

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

\begin{frame}
  \frametitle{Question}\label{question}
  \begin{itemize}
  \item How much of part and full-time employment is driven by
    \begin{itemize}
    \item aggregate shocks?
    \item match quality composition.
    \end{itemize}
  \item What are the consequences of ignoring part-time employment?
  \item Policy analysis
    \begin{itemize}
    \item UI vs. job-keeper
    \end{itemize}
  \item Today: first part.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{What We Do}\label{whatido}
  \begin{itemize}
  \item Develop a search and matching model of the labor market with
    \begin{itemize}
    \item endogenous part and full-time employment.
    \item endogenous transitions (PT/FT and to unemployment).
    \end{itemize}
  \item Mortensen-Pissarides (1994) with
    \begin{itemize}
    \item procyclical productivity that is higher for full-time jobs;
    \item acyclical costs that are asymmetric between part and full-time jobs;
    \item aggregate shocks.
    \end{itemize}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Preview of Findings}\label{question}
  \begin{itemize}
  \item How much of part and full-time employment is driven by
    \begin{itemize}
    \item aggregate shocks?
      \begin{itemize}
      \item Primarily drives job-finding.
      \end{itemize}
    \item match quality composition.
      \begin{itemize}
      \item Drives almost all fluctuations in part and full-time.
      \end{itemize}
    \end{itemize}
  \item What are the consequences of ignoring part-time employment?
    \begin{itemize}
    \item Understates the cost of business cycles \& magnitudes.
    \end{itemize}
  \item Policy analysis
    \begin{itemize}
    \item UI vs. job-keeper
      \begin{itemize}
      \item Job keeper very effective at limiting size of recession.
      \end{itemize}
    \end{itemize}
  \end{itemize}
\end{frame}

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

% \begin{frame}
%   \frametitle{Outline}\label{outline}
%   \begin{itemize}
%   \item Model
%   \item Simulated Method of Moments
%   \item Findings
%   \item Experiments
%   \item Conclude
%   \end{itemize}
% \end{frame}

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

\section{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{Model Environment}\label{environment}
  \begin{itemize}
  \item Random search and matching with endogenous separations.
  \item Continuous time, discount rate $r$.
  \item Agents:
    \begin{itemize}
    \item Unemployed and employed workers.
    \item Matched and unmatched firms.
  \end{itemize}
  \item Technology:
    \begin{itemize}
    \item Random matching in labor markets.
    \item Production: $z{\color{red}Y_{T}}\epsilon$ (agg, type, idiosyncratic).
    \item Endogenous transitions: between emp. types \& unemp.
    \end{itemize}
  \item For simplicity: assume agg. productivity ($z$) fixed (for now).
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Agents}\label{environment}
  \begin{itemize}
  \item Workers:
    \begin{itemize}
    \item May be unemployed, or employed part or full-time.
    \item Nash bargained wages, i.e., share of current match surplus.
  \end{itemize}
  \item Firms:
    \begin{itemize}
    \item Post single-worker vacancies at cost $\kappa$.
    \item Pay wages and costs depending on part or full-time worker.
    \item {\color{red}Costs: $\tau_{F}$ and $\tau_{P}$ for part and full-time.}
    \end{itemize}
  \item Jointly decide if match is full-time, part-time, or separate any time a shock occurs.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Search and Matching Technology}\label{searchandmatching}
  \begin{itemize}
  \item Random matching w/ sep. (Mortensen and Pissarides, 1994):
    \begin{itemize}
    \item Matches random: productivity $\epsilon$ not known before contact.
    \item Matches separate if $\epsilon$ is/falls below threshold (here $\epsilon_{P}$)
    \end{itemize}
  \item No OTJS.
  \item Matching technology:
    \begin{itemize}
    \item \# of matches in labor market: $M = M(u,v)$ (CRS).
    % \item Definitions:
    %   \begin{itemize}
      \item Labor Market Tightness: $\theta(\cdot) = \frac{v}{u}$
      \item Worker finding rate: $q(\theta) = \frac{M(u,v)}{v}$
      \item Job finding rates: $p(\theta) = \frac{M(u,v)}{u} = \theta q(\theta)$
      % \end{itemize}
    \end{itemize}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Workers}\label{firms}
  \begin{itemize}
  \item Either part-time or full-time ($T = \{P, F\}$).
  \item iid productivity: draw $\epsilon\sim_{iid}F(\epsilon)$; evolve at rate $\lambda_{T}$
  \item Wages determined by Nash Bargaining (bargaining power $\alpha$).
  \item Value of unemployment:
\begin{equation*}
  \label{eq:U}
  r\;U = b + p(\theta)\int_{\underline{\epsilon}}^{\bar{\epsilon}}[\max\{{\color{red}\max\{W^{F}(x),W^{P}(x)}\},0\} - U]dF(x).
\end{equation*}
\item Value of employment:
\begin{align*}
  \label{eq:W}\small
  \hspace{-10mm}   r\; W^{T}(\epsilon) = w + \lambda_{T}\alpha\int_{\underline{\epsilon}}^{\bar{\epsilon}}[\max\{{\color{red}\max\{S^{F}(x),S^{P}(x)\}},0\} - S^{T}(\epsilon,H)] \; dF(x)
\end{align*}
\item $S^{T}(x)$: joint surplus of firm \& worker.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Firms}\label{freeentry}
  \begin{itemize}
  \item Post vacancy at cost $\kappa$.
  \item Pay flow cost $\tau_{T}$ by type once employed.
    \item Value of a filled vacancy:
\begin{align*}\tiny
  \label{eq:J}
\hspace{-15mm}  r\; J^{T}(&\epsilon) = zY_{T}\epsilon - {\color{red}\tau_{T}} - w \\&+ \lambda_{T}(1 - \alpha)\int_{\underline{\epsilon}}^{\bar{\epsilon}}[\max\{{\color{red}\max\{S^{F}(x),S^{P}(x)\}},0\} - S^{T}(\epsilon)] \; dF(x)
\end{align*}
    \item Value of unfilled vacancy:
\begin{equation*}
  \label{eq:V}
  r\; V = -\kappa + q(\theta)\int_{\underline{\epsilon}}^{\bar{\epsilon}}[\max\{{\color{red}\max\{J^{F}(x),J^{P}(x)}\},0\} - V]dF(x)
\end{equation*}
\item Free entry ($V=0$)$\rightarrow$ match rate:   $q(\theta) = \frac{\kappa}{\int_{\epsilon_{P}} J(x,H)dF(x)}$
\item Market tightness: $\theta = q^{-1}(\frac{\kappa}{\int J dF(x)})$
  \end{itemize}
\end{frame}


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



\begin{frame}
  \frametitle{Surplus and Employment Thresholds}\label{freeentry}
  \begin{itemize}
  \item Surplus $S^{T}(\epsilon) &= W^{T}(\epsilon) - U + J^{T}(\epsilon) - V$
    \item Surplus of match for either $T = \{P, F\}$:
\begin{align*}
        (r + \lambda_{T})\; S^{T}(\epsilon) &= z\epsilon Y_{T} - \tau_{T} - b - \frac{\alpha}{1 - \alpha}\theta\kappa \nonumber\\&+ \lambda_{T}[\int_{\epsilon_{F}}^{\bar{\epsilon}}S^{F}(x)dF(x) + \int_{\epsilon_{P}}^{\epsilon_{F}}S^{P}(x)dF(x)]
\end{align*}
    \item Existence: assume $\exists$ some $\epsilon_{F}$ and $\epsilon_{P}$ st
      \begin{enumerate}
      \item $zY_{F}\epsilon_{F} - \tau_{F} > zY_{P}\epsilon_{F} - \tau_{P}$ and
      \item $zY_{P}\epsilon_{P} - \tau_{P} > 0$:
\end{enumerate}\vspace{-2mm}
  \end{itemize}
\end{frame}


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

\begin{frame}
  \frametitle{Employment Thresholds}\label{surplus}
\begin{tikzpicture}[scale=1.1]

% Axis

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

\node [left] at (0,5.3) {$S^{T}(\epsilon)$};

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

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

\draw [thick] (0,0.5) to (5,5.5);

% Upward Slopping P'Q'

\draw [thick] (0,1.5) to (5,4);

\draw [thick] (0,2) to (5,2);

\node [below] at (5.5,4) {$S(\epsilon,P)$};

\node [below] at (5.5,5.5) {$S^{F}(\epsilon)$};

\node [below] at (5.5,2) {$S^{P}(\epsilon_{P})=0$};


% dashed lines

\draw [dashed] (2,2.4)--(2,0);

\node [below] at (2,0) {$\epsilon_{F}$};

\draw [dashed] (1,2)--(1,0);

\node [below] at (1,0) {$\epsilon_{P}$};
\node [above] at (0.5,1) {$u$};

\node [above] at (1.5,1) {$e_{P}$};

\node [above] at (3.5,1) {$e_{F}$};

\end{tikzpicture}
% \begin{tikzpicture}
%   \begin{scope}
%     \clip (2,2) circle (3cm);
%     \draw[fill=green]
%       (current bounding box.south west) --
%       (0.3,-0.5)..controls(2.55,2)..(0.3,4.5)
%       -- (current bounding box.north west) -- cycle;
%     \draw[] (3.7,-0.5)..controls(1.5,2)..(3.7,4.5);
%     \draw (2,5)--(2,-1);
%     \draw[fill] (2,2) circle [radius=0.1];
%   \end{scope}
%   \draw (2,2) circle (3cm);
% \end{tikzpicture}
  % \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Policy Functions}\label{freeentry}
  \begin{itemize}
  \item Super messy for different $\lambda_{T}$. Show for $\lambda_{T} = \lambda$.
  \item The part-time treshold is given by the indifference equality: $S^{P}(\epsilon_{P}) = 0$:
\begin{align*}
 \epsilon_{P} &=  \frac{\tau_{P} + b + \frac{\alpha}{1 - \alpha}\theta(z)\kappa}{zY_{P}} \\&- \frac{\frac{\lambda z Y_{F}}{r + \lambda}\left(\int_{\epsilon_{F}}^{\bar{\epsilon}}[1 - F(x)] \; dx + \frac{Y_{P}}{Y_{F}}\int_{\epsilon_{P}}^{\epsilon_{F}}[1 - F(x)] \; dx\right)}{zY_{P}}\\
\end{align*}
\item And the full-time threshold is given by the indifference condition $S^{F}(\epsilon_{F}) = S^{P}(\epsilon_{F})$
\begin{align*}
\epsilon_{F}  &= \frac{\tau_{F} - \tau_{P}}{z(Y_{F}-Y_{P})}.
\end{align*}
\item $W^{*} = \alpha S$ and $\theta^{*} = q^{-1}(\frac{\kappa}{(1 - \alpha)S})$ determined by SS surplus.
  \end{itemize}
\end{frame}


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

\begin{frame}
  \frametitle{Steady-State Employment}\label{freeentry}
  \begin{itemize}
  \item Flows:
    \begin{enumerate}
    \item $u\rightarrow PT$, $u\rightarrow FT$;
    \item $PT\rightarrow FT$, $PT\rightarrow u$;
    \item $FT\rightarrow PT$, $FT\rightarrow u$.
    \end{enumerate}
  \item Must all be equal to 0 in equilibrium.
  \item Steady-state employment:
\begin{align*}
  e^{P} &= \frac{(P(\theta)u + \lambda_{F} e^{F})\; [F(\epsilon_{F}) - F(\epsilon_{P})]}{(\lambda_{P}[1 - F(\epsilon_{F}) + F(\epsilon_{P})])};\\
  e^{F} &= \frac{(P(\theta)u + \lambda_{P} e^{P})\; [1 - F(\epsilon_{F})]}{(\lambda_{F} F(\epsilon_{F}))};\\
u &= \frac{\lambda_{P} F(\epsilon_{P})e^{P} + \lambda_{F} F(\epsilon_{P})e^{F}}{p(\theta)[1 - F(\epsilon_{P})]}.
  % u &= \frac{\lambda F(\epsilon_{P})}{F(\epsilon_{P})[p(\theta) + \lambda]}.
\end{align*}
  \end{itemize}
\end{frame}


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

\begin{frame}
  \frametitle{Steady-State Equilibrium}\label{freeentry}
  \begin{itemize}
  \item Equilibrium is defined by the functions $(\theta^{*},w^{*},\epsilon_{F},\epsilon_{P})$, corresponding value functions, the steady-state tuple: $(e_{P},e_{F},u)$ such that
    \begin{enumerate}
    \item $\theta$ is defined by the free entry condition, $V = 0$, given other equilibrium objects, $\theta^{*} = q^{-1}(\frac{\kappa}{(1 - \alpha)S})$.
    \item $\epsilon_{P}$ defines the point at which $S^{P}(\epsilon_{P}) = 0$.
    \item $\epsilon_{F}$ defines the point at which $S^{F}(\epsilon_{F}) = S^{P}(\epsilon_{F})$
    \item Wages are given by a surplus sharing rule, $w^{T}(\epsilon) = \alpha S^{T}(\epsilon)$
    \end{enumerate}
  \end{itemize}
\end{frame}


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



% \begin{frame}
%   \frametitle{Comparative Statics: Unemployment}\label{surplus}
%   \begin{itemize}
%   \item Steady-state unemployment:
%     \begin{align*}
%   u &= \frac{\delta + \lambda F(\epsilon_{P})}{F(\epsilon_{P})[p(\theta(z)) + \lambda] + \delta}.
%     \end{align*}
%   \item Cutoffs:
% \begin{align*}
%  \epsilon_{P} &=  \frac{\tau_{P} + b + \frac{\alpha}{1 - \alpha}\theta(z)\kappa}{zY_{P}} \\&- \frac{\frac{\lambda z Y_{F}}{r + \lambda}\left(\int_{\epsilon_{F}}^{\bar{\epsilon}}[1 - F(x)] \; dx + \frac{Y_{P}}{Y_{F}}\int_{\epsilon_{P}}^{\epsilon_{F}}[1 - F(x)] \; dx\right)}{zY_{P}}\\
% % \end{align*}
% % \begin{align*}
% \epsilon_{F}  &= \frac{\tau_{F} - \tau_{P}}{z(Y_{F}-Y_{P})}.
% \end{align*}
%   \end{itemize}
% \end{frame}

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

% \begin{frame}
%   \frametitle{Intuition for Dynamics}\label{surplus}
%   \begin{itemize}
%   \item Dynamics determined by surplus of a match.
%   \item Surplus:
%     \begin{align*}
%       S(\epsilon, H) = J(\epsilon, H) + W(\epsilon, H) - U - \cancel{V}
%     \end{align*}
%   \item Nash bargaining:
% \begin{align*}
% \label{eq:w_solved}
% \alpha S(\epsilon,H) &= W(\epsilon, H) - U\\
% (1 - \alpha)S(\epsilon,H) &= J(\epsilon, H)
% \end{align*}
% \item Vacancies determined by surplus:
%   \begin{align*}
%     q(\theta) &= \frac{\kappa}{\int_{\epsilon_{P}}^{\bar{\epsilon}}J(x,H(x))} (\text{free entry})\\
%     q(\theta) &= \frac{\kappa}{(1 - \alpha)\int_{\epsilon_{P}}^{\bar{\epsilon}}S(x,H(x))}\\
%     \theta &= q^{-1}(\frac{\kappa}{(1 - \alpha)\int_{\epsilon_{P}}^{\bar{\epsilon}}S(x,H(x))})\\
%   \end{align*}
%   \end{itemize}
% \end{frame}

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

% \begin{frame}
%   \frametitle{Comparative Statics: Vacancies II}\label{surplus}
%   \begin{itemize}
% \item Vacancies determined by surplus:
%   \begin{align*}
%     q(\theta) &= \frac{\kappa}{(1 - \alpha)S(\epsilon,H)}\\
%     \theta &= q^{-1}(\frac{\kappa}{(1 - \alpha)S(\epsilon,H)})\\
%   \end{align*}
% \begin{align*}
% %    (r + \lambda) \; S(z, \epsilon, H) &= zY_{T}\epsilon - \tau_{T} + \lambda\int(\max\{S(z,x,H'),0\}) \; dF(x) - b - \frac{\alpha}{1 - \alpha}\theta(z)\kappa \\
%     (r + \lambda) \; S(z, \epsilon, H) &= zY_{T}\epsilon - \tau_{T} \\&+ \frac{\lambda z Y_{F}}{r + \lambda}\left(\int_{\epsilon_{F}}^{\bar{\epsilon}}[1 - F(x)]\; dx \\&+ \frac{Y_{P}}{Y_{F}}\int_{\epsilon_{P}}^{\epsilon_{F}}[1 - F(x)] \; dx ) - b - \frac{\alpha}{1 - \alpha}\theta(z)\kappa.
% \end{align*}
%   \end{itemize}
% \end{frame}

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


\begin{frame}
  \frametitle{Employment Threshold Dynamics}\label{surplus}
  \begin{itemize}
  \item Suppose $z\downarrow$
  \begin{tikzpicture}[scale=1.1,
    blacknode/.style={shape=circle, draw=black, line width=2},
    bluenode/.style={shape=circle, draw=blue!50!white, line width=2},
    greennode/.style={shape=circle, draw=green!50!white, line width=2},
    rednode/.style={shape=circle, draw=red!50!white, line width=2}
]

% Axis

\draw[fill=red!50!white] (1,2)--(2,2.5)--(2,0)--(1,0);
% \draw[thick,pattern=north east line,pattern color=blue!50!white] (2,2.5) -- (2.5,2.96) -- (2.5,0) -- (2,0);
\draw[fill=blue!50!white] (2,2.5) -- (2.5,2.96) -- (2.5,0) -- (2,0);
\draw[fill=green!50!white] (2.5,2.96)--(3.4,2.7)--(3.4,0)--(2.5,0);



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

\node [left] at (0,5.3) {$S^{T}(\epsilon)$};

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

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

\draw [dashed] (0,0.5) to (5,5.5);

% Upward Slopping P'Q'

\draw [dashed] (0,1.5) to (5,4);

\draw [dashed] (0,2) to (5,2);

\node [below] at (5.5,4) {$S^{P}(\epsilon; z)$};

\node [below] at (5.5,5.5) {$S^{F}(\epsilon; z)$};

\node [below] at (5.5,2) {$S^{P}(\epsilon_{P}; z)=0$};

\draw [thick,red] (0,0) to (5,4);

\draw [thick,red] (0,1) to (5,3.5);

\draw [thick,red] (0,2.25) to (5,2.25);

\node [above] at (5.5,4) {$S^{F}(\epsilon; z')$};

\node [below] at (5.5,3.5) {$S^{P}(\epsilon; z')$};

\node [above] at (5.5,2.25) {$S^{P}(\epsilon_{P}; z')=0$};

% dashed lines

\draw [dashed] (2.5,2.2)--(2.5,0);

% \node [below] at (2,0) {$\epsilon_{F}$};
\node [below] at (2.5,0) {$\epsilon_{P}(z')$};

\draw [dashed] (3.4,2.7)--(3.4,0);

\node [below] at (3.4,0) {$\epsilon_{F}(z')$};
\node [above] at (1.5,1) {$u'$};

\node [above] at (3,1) {$e_{P}'$};

\node [above] at (4.5,1) {$e_{F}'$};

\matrix [draw,below left] at (2,5.5) {
  \node [rednode,label=right:PT-U] {}; \\
  \node [bluenode,label=right:FT-U] {}; \\
  \node [greennode,label=right:FT-PT] {}; \\
};

\end{tikzpicture}
% \begin{tikzpicture}
%   \begin{scope}
%     \clip (2,2) circle (3cm);
%     \draw[fill=green]
%       (current bounding box.south west) --
%       (0.3,-0.5)..controls(2.55,2)..(0.3,4.5)
%       -- (current bounding box.north west) -- cycle;
%     \draw[] (3.7,-0.5)..controls(1.5,2)..(3.7,4.5);
%     \draw (2,5)--(2,-1);
%     \draw[fill] (2,2) circle [radius=0.1];
%   \end{scope}
%   \draw (2,2) circle (3cm);
% \end{tikzpicture}
  \end{itemize}
\end{frame}

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

% \begin{frame}
%   \frametitle{Acyclical Costs and Surplus}\label{surplus}
%   \begin{itemize}
%   \item Surplus at part-time cutoff: $S(\epsilon_{P},P) = 0$:
%     \begin{align*}
%       S(\epsilon,H(\epsilon)) = S(\epsilon,H(\epsilon)) &- S(\epsilon_{P},P)
% \end{align*}
% \begin{align*}\hspace{-10mm}
% % \rightarrow\int S(x,H(x))dF(x) &= \int [S(x,H(x)) - S(\epsilon_{P},P)]dF(x) \\
% \rightarrow\int S(x,H(x))dF(x) &= \int_{\epsilon_{F}}^{\bar{\epsilon}}\underbrace{\frac{zY_{F}x - \tau_{F}}{r + \lambda}}_{FT\;Surp}dF(x) + \int_{\epsilon_{P}}^{\epsilon_{F}}\underbrace{\frac{zY_{P}x - \tau_{P}}{r + \lambda}}_{PT\;Surp}dF(x)\\& - [1 - F(\epsilon_{P})]\frac{zY_{P}\epsilon_{P} - \tau_{P}}{r + \lambda}
%     \end{align*}
%   \item Acylical costs asymmetrically affect FT and PT surplus wrt $z$.
% \item What does this mean:
%   \begin{enumerate}
%   \item $\tau_{T}$: causes procyclical fluctuation in $\epsilon_{T}$.
%   \item Vacancy volatility increases with cost increases.
%   \end{enumerate}
%   \end{itemize}
% \end{frame}

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

% \begin{frame}
%   \frametitle{Comparative Statics II}\label{surplus}
%   \begin{itemize}
%   \item Surplus at part-time cutoff: $S(\epsilon_{P},P) = 0$:
%     \begin{align}
%       S(\epsilon,H(\epsilon)) - S(\epsilon_{P},P) &= S(\epsilon,H(\epsilon))
%       \int [S(x,H(x)) - S(\epsilon_{P},P)]dF(x) &= \int S(x,H(x))dF(x)
%     \end{align}
% \item Surplus in Mortensen-Pissarides (1994):
% \begin{align*}
%   \label{eq:SurpMP}\hspace{-5mm}
% (r + \lambda)S^{MP}(\epsilon) = zY\epsilon - b - \frac{\alpha}{1 - \alpha}\kappa\theta + \frac{\lambda zY}{r + \lambda}\int_{\epsilon_{D}}^{\bar{\epsilon}}[1 - F(x)]dx
% \end{align*}
% \item Surplus here by type:
% \begin{align*}
%     &\hspace{-5mm}(r + \lambda + \pi(H))S(\epsilon, H) = zY(H)\epsilon - b - {\color{red}\tau(H)} - \frac{\gamma(H)\alpha}{1 - \gamma(H)\alpha}\kappa\theta(H) \\ &\hspace{-5mm}+ \frac{\lambda zY(H)}{r + \lambda}\int_{{\color{blue}\epsilon_{D}^{H}}}^{\bar{\epsilon}}[1 - F(x)]dx + {\color{red}\pi(H) S(\epsilon, H')} - {\color{red}\delta_{T}(H)[U(H')-U(H)]}
% \end{align*}
% \item Market tightness:
%   \begin{align*}
%     q(\theta) &= \frac{\kappa}{(1 - \gamma(H)\alpha)S(\epsilon_{U},H)}\\
%     S(\epsilon_{D}^{H},H) &= 0\rightarrow S(\epsilon_{U},H) - S(\epsilon_{D}^{H},H) = S(\epsilon_{U},H)\\
%     \rightarrow q(\theta(H)) &= \frac{\kappa}{1 - \gamma(H)\alpha}\frac{r + \lambda + {\color{red}\pi(H)}}{zY(H)(\epsilon_{U} - {\color{blue}\epsilon_{D}^{H}})}
%   \end{align*}
%   \end{itemize}
% \end{frame}

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


% \begin{frame}
%   \frametitle{Comparative Statics III}\label{surplus}
%   \begin{itemize}
% \item Market tightness:
%   \begin{align*}
%     q(\theta) &= \frac{\kappa}{(1 - \alpha)\int_{\epsilon_{P}}^{\bar{\epsilon}}S(x,H(x))dF(x)}
% % \int S(x,H(x))dF(x) &= \int_{\epsilon_{F}}^{\bar{\epsilon}}\frac{zY_{F}x - \tau_{F}}{r + \lambda}dF(x) + \int_{\epsilon_{P}}^{\epsilon_{F}}\underbrace{\frac{zY_{P}x - \tau_{P}}{r + \lambda}}_{PT\;Surp}dF(x)\\& - [1 - F(\epsilon_{P})]\frac{zY_{P}\epsilon_{P} - \tau_{P}}{r + \lambda}
%   \end{align*}
% \item $\epsilon_{U}$: expected productivity from opening match.
% \item $\epsilon_{D}^{H}$: the least productive profitable match.
% \item What does this mean:
%   \begin{enumerate}
%   \item $\tau(H)$: causes procyclical fluctuation in $\epsilon_{D}^{H}$.
%   \item $\gamma(H)\alpha$: firm retains more surplus $\rightarrow$ compensated for costs.
%   \item $\pi(H)$: volatility in FT can increase volatility in PT.
%   \end{enumerate}
%   \end{itemize}
% \end{frame}

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

\section{Simulated Method of Moments} % 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{Calibration}\label{freeentry}
  \begin{itemize}
  \item Discretize model at weekly frequency.
  \item Preset parameters to ubiquitous values in literature.
  \item Estimate parameters related to novel features of the model.
  \item Target:
    \begin{itemize}
    \item Steady state employment rates;
    \item Steady state employment flows.
    \end{itemize}
  \item Simulate model 1000 times, with length of 320 quarters each.
  \item Average over 1000 simulations, toss first 200 quarters.
  \end{itemize}
\end{frame}

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

% \begin{frame}
%   \frametitle{Moments}\label{Moments}
%   \begin{itemize}
%   \item Parameters to estimate: $Y_{P}$, $Y_{F}$, $\tau_{P}$, $\tau_{F}$, $\lambda_{P}$, $\lambda_{F}$, $A$.
%   \item These parameters:
%     \begin{itemize}
%     \item Control the relative surplus of part and full-time employment, as well as unemployment.
%     \item i.e., employment rates.
%     \end{itemize}
%   \item Use CPS (1994-2018) to match:
%     \begin{itemize}
%     \item Average part-time, full-time, and unemployment rates.
%     \item Flows between employment states.
%   \end{itemize}
% \item Formal proof (check on your own): \hyperlink{equivalence}{\beamergotobutton{formal proof}}
%   \end{itemize}
% \end{frame}

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

\section{Findings}

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

\begin{frame}
  \frametitle{Outline for Findings}\label{Findings}
  \begin{itemize}
  \item Describe how model fits data.
  \item Decompose into match quality and aggregate shocks.
  % \item Policy experiments:
  %   \begin{itemize}
  %   \item UI vs. ``job-keeper''
  %   \end{itemize}
  \end{itemize}
\end{frame}


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

\begin{frame}\frametitle{Employment Rates}
    \begin{figure}[ht]
        \begin{minipage}[b]{0.465\linewidth}
            \centering
              \includegraphics[width=0.8\linewidth]{./FTEmpRateDataCompGreatRecession.png}
            \caption{Full-Time}
            \label{fig:a}
        \end{minipage}
        \hspace{0.5cm}
        \begin{minipage}[b]{0.45\linewidth}
            \centering
            \includegraphics[width=0.8\linewidth]{./PTEmpRateDataCompGreatRecession.png}
            \caption{Part-Time}
            \label{fig:b}
        \end{minipage}
    \end{figure}
\end{frame}

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

\begin{frame}\frametitle{Employment Flows}
    \begin{figure}[ht]
        \begin{minipage}[b]{0.465\linewidth}
            \centering
            \includegraphics[width=0.8\linewidth]{./FTPTRateDataCompGreatRecession.png}
            \caption{Full-time to part-time}
            \label{fig:a}
        \end{minipage}
        \hspace{0.5cm}
        \begin{minipage}[b]{0.45\linewidth}
            \centering
            \includegraphics[width=0.8\linewidth]{./PTFTRateDataCompGreatRecession.png}
            \caption{Part-time to full-time}
            \label{fig:b}
        \end{minipage}
    \end{figure}
\end{frame}

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


\begin{frame}
  \frametitle{Findings}\label{Findings}
  \begin{itemize}
  \item Decompose into contribution of match quality and aggregate shocks:
    \begin{itemize}
    \item Fix share of part and full-time to steady-state values.
    \item Set $\delta_{t} = \frac{e_{t}\rightarrow u_{t + 1}}{e_{t}}$, and $p(\theta_{t}) = \frac{u_{t}\rightarrow e_{t + 1}}{u_{t}}$
    \end{itemize}
  \item Fluctuations in pt and ft driven by aggregate shocks. ie within variation.
  \end{itemize}
\end{frame}


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

\begin{frame}\frametitle{Employment}
  \begin{itemize}
  \item Part-time employment too volatile.
  \item Employers hoard part-time workers instead of firing them.
  \end{itemize}
    \begin{figure}[ht]
        \begin{minipage}[b]{0.465\linewidth}
            \centering
            \includegraphics[width=0.8\linewidth]{./EEDataCompGreatRecessionDecompDecomp_FTDiff.png}
            \caption{Full-Time}
            \label{fig:a}
        \end{minipage}
        \hspace{0.5cm}
        \begin{minipage}[b]{0.45\linewidth}
            \centering
            \includegraphics[width=0.8\linewidth]{./EEDataCompGreatRecessionDecompDecomp_PTDiff.png}
            \caption{Part-Time}
            \label{fig:b}
        \end{minipage}
    \end{figure}
\end{frame}

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

\begin{frame}\frametitle{Wages and output}
    \begin{figure}[ht]
        \begin{minipage}[b]{0.465\linewidth}
            \centering
            \includegraphics[width=0.8\linewidth]{./WWDataCompGreatRecessionDecompDecomp_AggDiff.png}
            \caption{Wages.}
            \label{fig:a}
        \end{minipage}
        \hspace{0.5cm}
        \begin{minipage}[b]{0.45\linewidth}
            \centering
            \includegraphics[width=0.8\linewidth]{./TotOutputDataCompGreatRecessionDecompDecomp_AggDiff.png}
            \caption{Aggregate output.}
            \label{fig:b}
        \end{minipage}
    \end{figure}
\end{frame}

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

% \section{Policy Experiments}

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

% \begin{frame}
%   \frametitle{Policy Experiments}\label{Policy_Experiments}
%   \begin{itemize}
%   \item Is UI the best way to prevent large recessions?
%   \item Could we exploit part-time employment instead?
%   \item Experiment: compare expansion of UI to an equally expensive part and full-time subsidy.
%   \end{itemize}
% \end{frame}


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

% \begin{frame}\frametitle{ACA Employment}
%     \begin{figure}[ht]
%         \begin{minipage}[b]{0.465\linewidth}
%             \centering
%             \includegraphics[width=\textwidth]{/comp_UIHoardingCompBaseline1CompMnth/EEUIHoardingCompBaseline1Comp_Agg.png}
%             \caption{Employment}
%             \label{fig:a}
%         \end{minipage}
%         \hspace{0.5cm}
%         \begin{minipage}[b]{0.45\linewidth}
%             \centering
%             \includegraphics[width=\textwidth]{/comp_UIHoardingCompBaseline1CompMnth/TotOutputUIHoardingCompBaseline1Comp_Agg.png}
%             \caption{Output}
%             \label{fig:b}
%         \end{minipage}
%     \end{figure}
% \end{frame}

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

\section{Conclusion}

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

 \begin{frame}
   \frametitle{Paper Findings Summary}\label{conclusion}
   \begin{itemize}
   \item Extend Mortensen and Pissarides (1994) to include
     \begin{itemize}
     \item two types of employment (part and full-time)
     \item different acyclical costs by employment type.
     \end{itemize}
   \item Run calibrated example.
   \item Findings:
     \begin{itemize}
     \item Match quality explains fluctuations in part and full-time employment.
     \item Job-keeper scheme more effective than UI.
     \end{itemize}
   \end{itemize}
 \end{frame}

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 \begin{frame}
   \frametitle{Next Time}
   \begin{itemize}
   \item Hosios condition!
  \item Reminder:
    \begin{itemize}
    \item HW6 due 5/8. ONLY Q1, not Q2!
    \end{itemize}
   \end{itemize}
 \end{frame}

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%  \begin{frame}
%    \frametitle{Thank You!}
% \centering Thank you!
%  \end{frame}

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

\appendix
\section{Additional Material}

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

\begin{frame}
  \frametitle{Preset-Parameters}\label{Findings}
  \begin{itemize}
  \item AR(1) aggregate shocks \& log-normal iid productivity:
    \begin{align}
      z_{t + 1} &= \rho z_{t} + \nu_{z}\\
      \nu_{z}&\sim LN(0,\sigma_{Z})\\
      \epsilon &\sim Ln(0,\sigma_{\epsilon})
    \end{align}
  \item Cobb-Douglas Matching: $M(u,v) = A u^{\eta}v^{1 - \eta}$.
  \item Preset parameters estimated in related papers.
\begin{table}[h]\label{tab:Test1_Results}
\centering\begin{center}
\resizebox{\textwidth}{!}{%
\begin{tabulary}{\textwidth}{L r l | l} \toprule \multicolumn{2}{c}{Parameter} & Value & Source \\
\midrule
    \multicolumn{2}{c}{$b$} & $\getvalround{Value}{paramsdata}{Param}{b}{1}$ & UI Percent of Output (Shimer, 2005) \\
    \multicolumn{2}{c}{$\eta$} & $\getvalround{Value}{paramsdata}{Param}{eeta}{1}$ & Matching function elasticity (Shimer, 2005) \\
    \multicolumn{2}{c}{$\alpha$} & $\getvalround{Value}{paramsdata}{Param}{aalpha}{1}$ & Hosios Condition \\
    \multicolumn{2}{c}{$A$} & $\getvalround{Value}{paramsdata}{Param}{A}{3}$ & Matching efficiency (Shimer, 2005) \\
    \multicolumn{2}{c}{$\sigma_{\epsilon}$} & $\getvalround{Value}{paramsdata}{Param}{SigEps}{2}$ & Variance of match productivity shock (Fujita and Ramey, 2012) \\
    \multicolumn{2}{c}{$\rho$} & $\getvalround{Value}{paramsdata}{Param}{RhoZ}{4}$ & Persistence of aggregate shocks (Fujita and Ramey, 2012) \\
    \multicolumn{2}{c}{$\sigma_{z}$} & $\getvalround{Value}{paramsdata}{Param}{SigZ}{3}$ & Variance of aggregate shocks (Fujita and Ramey, 2012) \\
    % \multicolumn{2}{c}{$\bar{\epsilon}$} & $\getvalround{Value}{paramsdata}{Param}{bar_epsilon}{2}$ & Average match idiosyncratic productivity \citep{FujitaRamey2012} \\
    % \multicolumn{2}{c}{$z$} & $\getvalround{Value}{paramsdata}{Param}{z}{2}$ & Average aggregate output \\
\bottomrule
\end{tabulary}
}
\end{center}
\end{table}
\hyperlink{calibration}{\beamerreturnbutton{back}}
  \end{itemize}
\end{frame}

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\begin{frame}
  \frametitle{Fit}\label{Moments}
  \begin{itemize}
  \item Matches moments well.
\begin{table}[h]\caption{Fit}\label{tab:Moment_Matching}
\centering\begin{center}
\resizebox{\textwidth}{!}{%
\begin{tabulary}{\textwidth}{l c | l} \toprule Moment & Data & Model \\
\midrule Wage Ratio (Full-Time to Part-Time) & $\getvalround{Value}{moments}{Stat}{WageRatio}{3}$ & $\getvalround{Value}{modelmoments}{Stat}{WageRatio}{3}$ \\
    Full-Time Employment Rate & $\getvalround{Value}{moments}{Stat}{EEFT}{3}$ & $\getvalround{Value}{modelmoments}{Stat}{EEFT}{3}$ \\
    Part-Time Employment Rate & $\getvalround{Value}{moments}{Stat}{EEPT}{3}$ & $\getvalround{Value}{modelmoments}{Stat}{EEPT}{3}$ \\
    Unemployment Rate & $\getvalround{Value}{moments}{Stat}{UU}{3}$ & $\getvalround{Value}{modelmoments}{Stat}{UU}{3}$ \\
    Separation Rate & $\getvalround{Value}{moments}{Stat}{EU}{3}$ & $\getvalround{Value}{modelmoments}{Stat}{EU}{3}$ \\
\bottomrule
\end{tabulary}
}
\end{center}
% \begin{tablenotes}[flushleft]
% Notes: The table presents the change in lifetime utility (equivalent variation) for a one standard deviation change in each of the listed variables. When a variable is changed, the other variables are left unchanged.\end{tablenotes}
\end{table}
\hyperlink{calibration}{\beamerreturnbutton{back}}
  \end{itemize}
\end{frame}

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\begin{frame}
  \frametitle{Estimated Parameters}\label{Findings}
  \begin{itemize}
\vspace{-5mm}
\begin{table}[h]\label{tab:Test1_Results}
\centering\begin{center}
\resizebox{\textwidth}{!}{%
\begin{tabulary}{\textwidth}{C l | l} \toprule Parameter & Value & Source \\ \midrule
% \midrule $Estimated\;Parameters$ & & & \\
$\tau_{P}$ & $\getvalround{Value}{paramsdata}{Param}{ttauP}{2}$ & Acyclical cost for part-time workers \\
$\tau_{F}$ & $\getvalround{Value}{paramsdata}{Param}{ttauF}{2}$ & Acyclical cost for full-time workers \\
$Y_{P}$ & $\getvalround{Value}{paramsdata}{Param}{YP}{2}$ & Part-time productivity \\
$Y_{F}$ & $\getvalround{Value}{paramsdata}{Param}{YF}{2}$ & Full-time productivity \\
\bottomrule
\end{tabulary}
}
\end{center}
\end{table}
  \item $Y_{F}/Y_{P} \approx 1.38$.
  \item FT vs. PT hours (data): 40 hrs. vs. 28 hrs ($\approx 1.43$).
  \item $\tau_{F}/\tau_{P} \approx \$3.18/hour$.
  \item FT vs. PT costs (data): 9.75/hr. vs. 3.2/hr. ($\approx \$3.05/hour$).
  \end{itemize}
\hyperlink{calibration}{\beamerreturnbutton{back}}
\end{frame}

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\end{document}