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\hypersetup{pdftitle={Lecture | u* = √uv}}

\begin{document}

\title{$u^* = \sqrt{uv}$}
\information{Pascal Michaillat, Emmanuel Saez}%
{April 2022}%
{https://www.pascalmichaillat.org/13.html}
\frame[plain]{\titlepage}

\begin{frame}
\frametitle{A formula for efficient unemployment, $u^*$}
\begin{itemize}
\item[\then] sufficient statistic for optimal stabilization policies
\begin{itemize}
\item monetary policy
\item fiscal policy
\item unemployment insurance
\end{itemize}
\item[\then] welfare-based measure of ``full employment''
\begin{itemize}
\item statutory target for US policymakers
\item Full Employment and Balanced Growth Act of 1978
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Existing measures of ``full employment''}
\includegraphics<1>[scale=0.26,page=1]{\pdf}%
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\begin{frame}
\frametitle{This paper: a simple welfare-based measure}
\includegraphics<1>[scale=0.26,page=5]{\pdf}%
\end{frame}

\begin{frame}
\heading{Theory}
\end{frame}

\begin{frame}
\frametitle{Workers}
\begin{enumerate}
\item share $u$ of labor force is unemployed
\begin{itemize}
\item no contribution to social welfare
\end{itemize}
\item share $v$ of labor force is recruiting
\begin{itemize}
	\item one worker per vacancy
	\item \# vacancies determined by Beveridge curve $v(u)$
\end{itemize}
\item share $1-(u+v)$ of labor force is producing
\begin{itemize}
\item production determines social welfare
\end{itemize}
\end{enumerate}
\end{frame}

\begin{frame}
\frametitle{US Beveridge curve $\approx$ hyperbola}
\includegraphics<1>[scale=0.26,page=6]{\pdf}%
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\end{frame}

\begin{frame}
\frametitle{Social planner's problem}
\begin{itemize}
\item minimize nonproduction $u+v$
\item subject to Beveridge curve $v = A/u$, or $uv = A$
\item solution by symmetry: $u^*=v^* = \sqrt{A} = \sqrt{uv}$
\item solution by first-order condition:
\begin{itemize}
\item minimize $u+A/u$, which is convex
\item first-order condition is necessary \& sufficient
\begin{equation*}
1-A/u^2 = 0 \implies u = \sqrt{A}
\end{equation*}
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Efficient unemployment rate}
\begin{itemize}
\item efficient unemployment rate: 
\begin{equation*}
u^* = \sqrt{uv}
\end{equation*}
\item economy is inefficiently tight when $u<u^*$ or 
\begin{equation*}
u < v
\end{equation*}
\item economy is inefficiently slack when $u>u^*$ or
\begin{equation*}
u > v
\end{equation*}
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Generalization [Michaillat \& Saez 2021]}
\begin{itemize}
\item home production per unemployed worker: $0 \to \al{\z}$
\item \# recruiters per vacancy: $1 \to \al{\k}$
\item Beveridge curve: $v = A/u \to v = A/u^{\al{\e}}$
\item efficient tightness: 
\begin{equation*}
\t^* = 1 \quad\to\quad \t^* = \frac{1-\al{\z}}{\al{\k\e}}
\end{equation*}
\item efficient unemployment rate:
\begin{equation*}
u^* = \sqrt{uv} \quad\to\quad u^* =\bp{\frac{\al{\k} \cdot \al{\e}}{1-\al{\z}}\cdot v\cdot u^{\al{\e}}}^{1/(1+\al{\e})}
\end{equation*}
\end{itemize}	
\end{frame}

\begin{frame}
\frametitle{Hagedorn \& Manovskii [2008]: $\z = 0.96$}
\includegraphics<1>[scale=0.26,page=49]{\pdf}%
\end{frame}

\begin{frame}
\frametitle{Limited influence of matching dynamics}
\begin{enumerate}
\item unemployment is almost always on Beveridge curve
\begin{itemize}
\item $\dot{u}(t) = \l \cdot [1-u(t)] - f \cdot u(t)$
\item Beveridge curve: $\dot{u}=0 \implies u^b = \l/(\l+f)$
\item unemployment dynamics: $u(t) - u^b = [u(0) - u^b] e^{\al{-(\l+f)t}}$
\item half life of $u(t) - u^b$: $\ln(2)/(\l+f) = \ln(2)/\al{0.59} = \al{1.17}$ \al{month}
\end{itemize}	
\item efficient tightness $\t^*$ is almost equal to Hosios tightness $\t^h$
\begin{itemize}
\item with interest rate $r$ and matching elasticity $\h$:
\begin{equation*}
\frac{\t^*-\t^h}{\t^*} = \frac{\al{r}}{\h (\l+f)}
\end{equation*}
\item under Shimer [2005] calibration: $(\t^*-\t^h)/\t^* = \al{1.1\%}$
\end{itemize}	
\end{enumerate}
\end{frame}

\begin{frame}
\frametitle{Unemployment $\approx$ on Beveridge curve}
\includegraphics<1>[scale=0.26,page=50]{\pdf}%
\includegraphics<2>[scale=0.26,page=51]{\pdf}%
\end{frame}

\begin{frame}
\frametitle{Efficient unemployment $\approx$ Hosios}
\includegraphics<1>[scale=0.26,page=52]{\pdf}%
\includegraphics<2>[scale=0.26,page=53]{\pdf}%
\end{frame}

\begin{frame}
\heading{United States, 1951--2019}
\end{frame}

\begin{frame}
\frametitle{Unemployment rate (CPS)}
\includegraphics[scale=0.26,page=12]{\pdf}%
\end{frame}

\begin{frame}
\frametitle{Vacancy rate (Barnichon 2010 \& JOLTS)}
\includegraphics[scale=0.26,page=13]{\pdf}%
\end{frame}

\begin{frame}
\frametitle{Economy is too slack when $u>v$}
\includegraphics[scale=0.26,page=14]{\pdf}%
\end{frame}

\begin{frame}
\frametitle{Economy is too tight when $u<v$}
\includegraphics[scale=0.26,page=15]{\pdf}%
\end{frame}

\begin{frame}
\frametitle{Efficient unemployment rate is $\sqrt{uv}$}
\includegraphics[scale=0.26,page=16]{\pdf}%
\end{frame}

\begin{frame}
\frametitle{Unemployment gap is countercyclical}
\includegraphics[scale=0.26,page=17]{\pdf}%
\end{frame}

\begin{frame}
\heading{United States, 2020--2022}
\end{frame}

\begin{frame}
\frametitle{Unemployment \& vacancy rates}
\includegraphics[scale=0.26,page=18]{\pdf}%
\end{frame}

\begin{frame}
\frametitle{Economy is too tight since May 2021}
\includegraphics[scale=0.26,page=19]{\pdf}%
\end{frame}

\begin{frame}
\frametitle{Efficient unemployment rate is $\sqrt{uv}$}
\includegraphics[scale=0.26,page=20]{\pdf}%
\end{frame}

\begin{frame}
\frametitle{Most extreme unemployment gaps on record}
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\end{frame}

\begin{frame}
\frametitle{Why is efficient unemployment now so high?}
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\end{document}