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%	TITLE PAGE
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\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 2025} % 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{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{Introduction}
  \begin{itemize}
  \item Today: the Hosios Condition
  \item Efficiency in search and matching models.
  \item (Note: largely derived from Christine Braun's lecture on the DMP model).
  \item Homework on my website.
  \item Code for Aiyagari w/ labor-leisure choice on the cluster.
  \end{itemize}
\end{frame}

% ------------------------------------------------
\section{Efficiency in the DMP Model}
% ------------------------------------------------

\begin{frame}
\frametitle{Efficiency}

\begin{itemize}
	\vfill \item Is zero unemployment efficient?  \alert{No}
	\begin{itemize}
		\vfill \item higher unemployment incentivizes firms to post vacancies
		\vfill \item but high unemployment is costly, less production
	\end{itemize}
	\vfill \item Is a high vacancy rate efficient?
	\begin{itemize}
		\vfill \item vacancy creation is costly
		\vfill \item but lots of vacancies reduces unemployment
	\end{itemize}
	\vfill \item So what is the efficient level of $\theta$?
\end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{Efficiency}

\begin{itemize}
	\vfill \item 	Congestion externality
	\begin{itemize}
		\vfill \item one more hiring firm makes unemployed workers better off and makes all other hiring firms worse off
		\vfill \item one more searching worker makes hiring firms better off and makes all other searching workers worse off
	\end{itemize}
	\vfill \item Appropriability
	\begin{itemize}
		\vfill \item firm pays a cost $\kappa$ to post vacancy but does not get to keep the entire output $p$
	\end{itemize}
\end{itemize}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Efficiency}

\begin{itemize}
	\vfill \item What value of $\theta$ would the social planer choose to maximize total output/utility if he is constrained by the same matching frictions?
		\begin{itemize}
			\vfill \item does not care about wage b/c it's a linear transfer from the firm to the worker
		\end{itemize}
	\vfill \item Does there exist a wage such that job creation is the same in the decentralized equilibrium as in the social planners outcome?
	\vfill \item Can we achieve this wage with the Nash solution?
\end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{The DMP Model (``Ch. 1 of Pissarides (2000)'')}
  \begin{itemize}
  \item Agents:
    \begin{enumerate}
    \item Employed workers;
    \item unemployed workers;
    \item vacant firms;
    \item matched firms.
    \end{enumerate}
  \item Linear utility ($u = b, u = w$) and production $y = p>b$.
  \item Matching function:
    \begin{enumerate}
    \item Constant returns to scale ($L$ is lab. force):
    \begin{align*}
      M(uL,vL) = uL\times M(1,\frac{v}{u}) = uL\times p(\theta)
    \end{align*}
  \item where $\theta = \frac{v}{u}$ is ``labor market tightness''
  \item Match rates:
    \begin{align*}
      \underbrace{p(\theta)}_{Worker}= \theta \underbrace{q(\theta)}_{Firm}
    \end{align*}
    \end{enumerate}
  \item Social planner: pick $\theta$ optimally, no need to respect free entry condition.
  \end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{Social Planner's Problem}

$$ \int_0^\infty e^{-rt}[p(1-u)+bu - \kappa \theta u]~dt$$
$$\text{s.t. } ~ \dot{u} = \delta(1-u) - p(\theta)u$$
\begin{itemize}
	\item Social planner's problem
        \begin{itemize}
        		\vfill \item $p(1-u)$: social output of employment
        		\vfill \item $bu$: leisure enjoyed by unemployed workers
        		\vfill \item  $\kappa \theta u$: cost of jobs
        \end{itemize}
        \vfill \item Social planner is subject to the same transition equation for unemployment
\end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Social Planner's Problem}


\begin{itemize}
	\vfill \item The Hamiltonian
		$$H = e^{-rt}[p(1-u) + bu - \kappa \theta u] + \mu(t)[\delta(1-u) - p(\theta)u]$$
	\vfill \item FOCs
		\begin{align*}
		&H_{u} = -\dot{\mu} + r\mu \Rightarrow &-e^{-rt}(p-b+\kappa \theta) - [\delta + r + p(\theta)]\mu + \dot{\mu} = 0\\
		&H_{\theta} = 0 \Rightarrow &-e^{-rt}\kappa u - \mu u (q(\theta) + \theta q'(\theta)) = 0
		\end{align*}
              \item $\mu$: marginal value of an extra unemployed worker.
\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Social Planner's Problem}


\begin{itemize}
	\vfill \item Optimal $\theta$
		\begin{align*}
		&H_{\theta} = 0 \Rightarrow &-e^{-rt}\kappa u - \mu u q(\theta)(1 + \frac{\theta q'(\theta)}{q(\theta)}) = 0
		\end{align*}
              \item What is $\frac{\theta q'(\theta)}{q(\theta)}$?
                \begin{align*}
                  m(u,v) &= v q(\theta)\\
                  \rightarrow \der{m(u,v)}{u} &= v q'(\theta)\frac{-v}{u^{2}}\\
                  \rightarrow \der{m(u,v)}{u} &= -\theta^{2} q'(\theta)\\
                  \rightarrow \frac{\der{m(u,v)}{u}}{m(u,v)} &= -\frac{\theta^{2} q'(\theta)}{vq(\theta)}\\
                  \rightarrow u\frac{\der{m(u,v)}{u}}{m(u,v)} &= -\frac{\theta q'(\theta)}{q(\theta)}
                \end{align*}
              \item $\frac{\theta q'(\theta)}{q(\theta)}$ is the elasticity of the matching function wrt u.
\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Social Planner's Problem}


\begin{itemize}
	\vfill \item The Hamiltonian
		$$H = e^{-rt}[p(1-u) + bu - \kappa \theta u] + \mu(t)[\delta(1-u) - p(\theta)u]$$
	\vfill \item FOCs
		\begin{align*}
		&H_{u} = -\dot{\mu}+r\mu \Rightarrow &-e^{-rt}(p-b+\kappa \theta) - [\delta + r + p(\theta)]\mu + \dot{\mu} = 0\\
		&H_{\theta} = 0 \Rightarrow &-e^{-rt}\kappa u - \mu u q(\theta)(1 - \eta(\theta)) = 0
		\end{align*}
              \item $\eta(\theta)$: elasticity of match fun. wrt u.
\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
\frametitle{Optimal $\theta$}

\begin{itemize}
	\vfill \item Using $p(\theta) = \theta q(\theta)$ and solving in steady state ($\dot{\mu} = 0$):
        \begin{align*}
          \frac{p - b + \kappa\theta}{\delta + r + p(\theta)} &= \frac{\kappa}{q(\theta)(1 - \eta(\theta))}\\
          (p - b)(1 - \eta(\theta)) +  \kappa(1 - \eta(\theta))\frac{p(\theta)}{q(\theta)} &= \frac{(\delta + r + p(\theta))\kappa}{q(\theta)}
        \end{align*}
	\begin{equation}\rightarrow (1-\alert{\eta(\theta)})(p - b) - \frac{\delta + r+ \alert{\eta(\theta)} p(\theta)}{q(\theta)} \kappa= 0\end{equation}
      \item This is optimal $\theta$
	% \vfill \item From the decentralized solution, plug the wage curve into the Job creation curve
	% \begin{equation}(1-\alert{\beta})(p-b) - \frac{\delta + r + \alert{\beta} p(\theta)}{q(\theta)} \kappa =0 \end{equation}
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Decentralized allocation}

\begin{frame}
  \frametitle{Decentralized solution}
  \begin{itemize}
  \item Can the decentralized solution achieve the same level of $\theta$?
  \item i.e., can the decentralized level of unemployment be \textit{efficient}?
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Decentralized $\theta$}
  \begin{itemize}
  \item Free entry $V = 0$:
    \begin{align*}
      rJ(w) &= (p - w) + \delta[\cancel{V} - J(w)]\\
      (r + \delta)J(w) &= (p - w)
    \end{align*}
  \item Vacancy creation condition (i.e., free entry imposed):
    \begin{align*}
      q(\theta) &= \frac{\kappa}{E[J(w)]} \\
      q(\theta) &= \frac{\kappa(r + \delta)}{(p - w)}\\
      \theta &= q^{-1}(\frac{\kappa(r + \delta)}{(p - w)})
    \end{align*}
  \item Thus, mapping between wages and $\theta$. 1 equation, 2 unknowns.
  \item Need equation to determine wages in equilibrium.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Wage Determination}
  \begin{itemize}
  \item Recall Nash Bargained wages:
    \begin{align*}
      w &= argmax_{w}\underbrace{(W(w) - U)^{\beta}}_{Net\;Utility}\underbrace{(J(w) - V)^{1 - \beta}}_{Net\;Profits}\\
      0 &= \beta (W(w) - U)^{\beta - 1}(J(w) - V)^{1 - \beta}\der{W}{w} \nonumber\\&+ (1 - \beta)(J(w) - V)^{-\beta}(W(w) - U)\der{J}{w}
    \end{align*}
  \item $\der{W}{w} = 1$, $\der{J}{w} = -1$:
    \begin{align*}
      \beta(\frac{J(w)}{W(w) - U})^{1 - \beta} &= (1 - \beta)(\frac{W(w) - U}{J(w)})^{\beta}\\
      \beta(J(w) + W(w) - U) &= W(w) - U\\
      \beta S(w) &= W(w) - U
    \end{align*}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Wage Determination}
  \begin{itemize}
  \item Note that $\beta S(w) = [W(w) - U]$
    \begin{align*}
      (1 - \beta)(w - b) &= \beta(p - w - \delta J(w)) \\&+ (1 - \beta)(p(\theta) + \delta)\beta S(w)
    \end{align*}
  \item And $(1 - \beta)S(w) = J(w)\rightarrow S(w) = \frac{J(w)}{1 - \beta}$
    \begin{align*}
      (1 - \beta)(w - b) &= \beta(p - w - \delta J(w)) \\&+ (1 - \beta)(p(\theta) + \delta)\beta\frac{J(w)}{1 - \beta}\\
      w &= (1 - \beta) b + \beta p + p(\theta)\beta J(w)
    \end{align*}
  \item Free entry condition: $q(\theta) = \frac{\kappa}{J(w)}\rightarrow p(\theta) = \frac{\theta\kappa}{J(w)}$
    \begin{align*}
      w &= (1 - \beta) b + \beta p + \beta\theta\kappa
    \end{align*}
  \end{itemize}
\end{frame}

% ------------------------------------------------
\section{Comparing decentralized and Planner's allocation}
% ------------------------------------------------

\begin{frame}
  \frametitle{Decentralized free entry}
  \begin{itemize}
  \item Job creation curve:
    \begin{align*}
      (r + \delta)J(w) &= (p - w)\\
      q(\theta) &= \frac{\kappa}{J(w)} \\
      q(\theta) &= \frac{\kappa(r + \delta)}{(p - w)} \\
      p - w - \frac{\kappa(r + \delta)}{q(\theta)} &= 0
    \end{align*}
  \item Now, plug in using wages we just found:
    \begin{align*}
      w &= (1 - \beta) b + \beta p + \beta\theta\kappa
    \end{align*}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Decentralized free entry}
  \begin{itemize}
  \item Job creation curve:
    \begin{align*}
      p - ((1 - \beta) b + \beta p + \beta\theta\kappa) - \frac{\kappa(r + \delta)}{q(\theta)} &= 0
    \end{align*}
  \item identities: $p(\theta) = \theta q(\theta)\rightarrow \theta = \frac{p(\theta)}{q(\theta)}$
    \begin{align*}
      \rightarrow p - ((1 - \beta) b + \beta p + \beta\frac{p(\theta)}{q(\theta)}\kappa) - \frac{\kappa(r + \delta)}{q(\theta)} &= 0\\
      (1 - \beta) (p - b) - \beta\frac{p(\theta)}{q(\theta)}\kappa) - \frac{\kappa(r + \delta)}{q(\theta)} &= 0\\
      (1 - \beta) (p - b) - \frac{r + \delta + \beta p(\theta)}{q(\theta)}\kappa &= 0
    \end{align*}
  \item Looks familiar?
  \end{itemize}
\end{frame}

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


\begin{frame}
\frametitle{Social Planner's Problem}

\begin{itemize}
	\vfill \item Using $p(\theta) = \theta q(\theta)$ and solving in steady state ($\dot{\mu} = 0$)
	\begin{equation}(1-\alert{\eta(\theta)})(p - b) - \frac{\delta + r+ \alert{\eta(\theta)} p(\theta)}{q(\theta)} \kappa= 0\end{equation}
	\vfill \item From the decentralized solution, plug the wage curve into the Job creation curve
	\begin{equation}(1-\alert{\beta})(p-b) - \frac{\delta + r + \alert{\beta} p(\theta)}{q(\theta)} \kappa =0 \end{equation}
\end{itemize}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Efficiency }


\begin{itemize}
	\vfill \item Comparing (1) and (2) we see that we have efficiency in the decentralized market if $\beta = \eta(\theta)$. The workers bargaining power is equal to the elasticity of the matching function with respect to $u$.
	\vfill \item This is a general result: we have efficiency when
	$$\eta(\theta) = \beta$$
	\vfill \item This is called the Hosios (1990) condition
\end{itemize}
\end{frame}

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

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

\begin{frame}
  \frametitle{A couple more classes}
  \begin{itemize}
  \item Two more classes, likely Directed Search and Block Recursive Equilibrium.
  \end{itemize}
\end{frame}


\begin{frame}
  \frametitle{Cats!}
\centering\includegraphics[width=0.5\textwidth]{./IMG_20200620_130234-COLLAGE.jpg}
\end{frame}

\end{document}
