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%	TITLE PAGE
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\title[]{Quantitative Macro-Labor:\\ Disciplining Wage Contracts with Convex Preferences} % 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 2024} % Date, can be changed to a custom date

\begin{document}

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  \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 Today: Extend Burdett-Mortensen to environment where firms post {\it wage-tenure contracts} instead of wages.
  \item Research Proposal/Introduction: Due next Thursday.
  \item Short presentations (probably) the following week.
  \item May instead start section on Market Power.
  \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}
  \item But, wages also increase with {\it tenure}.
  \item BM model can't account for this.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{The Burdett-Mortensen OTJS 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 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 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 {\bf Firms cannot respond to outside offers.}
  \item Are these contracts optimal?
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{The Burdett-Mortensen Model}
  \begin{itemize}
  \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 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{The Firm}
  \begin{itemize}
  \item Assume equilibrium conditions \& $\lambda = \alpha$.
  \item Define $\pi^{V}$ as the profits of a vacant firm.
  \item Firm profit function:
    \begin{align}
      \pi^{V} &= \max_{w} (p - w)l(w|w_{R}, F)\\
      \pi^{V}(w|w_{R},F) &= (p - w)l(w|w_{R}, F)\\
      l(w|w_{R}, F) &= \frac{m\alpha\delta}{(\delta + \alpha[1 - F(w)])^{2}}
    \end{align}
  \item What is $l$? It is the probability of meeting a workers {\it and} the expected duration a worker employed at wage $w$ will be employed with a firm.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Matched Firm Value}
  \begin{itemize}
  \item Matched firm profit function:
    \begin{align}
      r\pi^{F}(w|w_{R},F) &= (p - w) + \alpha[1 - F(w)](\pi^{V} - \pi^{F}(w|w_{R},F)) \nonumber\\&+ \delta(\pi^{V} - \pi^{F}(w|w_{R},F))\\
\end{align}
\item Would a matched firm prefer to retain a worker?
\begin{align}
      r\pi^{F}(w_{R}|w_{R},F) &= (p - w_{R}) + \alpha[1 - \cancel{F(w_{R})}](\pi^{V} - \pi^{F}(w_{R}|w_{R},F)) \nonumber\\&+ \delta(\pi^{V} - \pi^{F}(w_{R}|w_{R},F))\\
      r\pi^{F}(w_{R}|w_{R},F) &= (p - w_{R}) + (\alpha + \delta)(\pi^{V} - \pi^{F}(w_{R}|w_{R},F))
\end{align}
\item $\pi^{F} \geq \pi^{V}$. Why is this?
\item The firm would prefer to retain a worker.
\item Maybe there is room for better contracts?
  \end{itemize}
\end{frame}

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

\section{Burdett and Coles (2003)}
% ------------------------------------------------

\begin{frame}
  \frametitle{Burdett and Coles (2003)}
  \begin{itemize}
  \item Another way of thinking about on-the-job search: moral hazard.
  \item Firm can't contract on worker behavior: worker can leave anytime it finds a higher wage.
  \item How could we allow a firm to handle this?
    \begin{enumerate}
    \item Respond to outside offers. (doesn't make sense in model with identical productivity).
    \item Write contracts that change wages to mitigate moral hazard.
    \end{enumerate}
  \item What does option 2 mean? They will increase wages the longer a worker remains employed.
  \item Backloaded contracts: option value larger so worker wants to stay.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Outline (Burdett-Coles, 2003)}
  \begin{itemize}
  \item Preferences and Technology:
    \begin{enumerate}
    \item Workers can be in one of two states: employed or unemployed, with value functions $V_{E}, V_{U}$.
    \item Firms post wage-{\it contracts}: distribution of {\it promised values},  $V\sim F(.)$.
    \item Workers receive job offers at exogenous rate $\lambda$.
    \item Separate two ways: ``exit the model'' (rate $\delta$) and via thru OTJS (rate $\lambda [1 - F(V)]$)
    \item Risk-aversion: $u(c) = u(b)$ or $u(c) = u(w)$.
    \item Rate of time preference: $r = 0$ (simplification for analytical results).
    \end{enumerate}
  \item {\bf Firms cannot respond to outside offers.}
  \item Instead, they increase wages over time to ensure that a worker does not leave early.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Contracts}
  \begin{itemize}
  \item Rather than wages, a firm offers a promised value, and can distribute it in any way over the life of the contract.
  \item Optimally, they want to backload the contract.
  \item Why? Because workers are less likely to leave if continuation value is high.
  \item Wage-tenure contracts:
    \begin{enumerate}
    \item Wage $w(.)\geq 0$.
    \item Where tenure, $t$, is its only argument.
    \item By assumption, equilibrium is symmetric, i.e., all contracts are the same given $t$.
    \end{enumerate}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Worker Value Functions}
  \begin{itemize}
  \item Unemployed flow value (bc $r = 0$):
    \begin{align}
      \delta V_{U} = u(b) + \lambda\int_{V_{U}}^{\bar{V}}[x - V_{U}]dF(x)
    \end{align}
  \item Employed value function
    \begin{align}
      \delta V_{E}(t|w(.)) - \frac{d V_{E}(t|w(.))}{dt} &= u(w(t)) \nonumber\\&+ \lambda\int_{V_{E}(t|w(.))}^{\bar{V}}[V_{x} - V_{E}(t|w(.))]dF(V_{x})
    \end{align}
  \item Currently, optimal contract unspecified.
  \item What is $\frac{d V_{E}(t|w(.))}{dt}$?
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Firm Value Functions}
  \begin{itemize}
  \item Define a worker survival probability (i.e., doesn't leave or die.) through tenure $t$.
    \begin{align}
      \psi(t|w(.)) = e^{-\int_{0}^{t}[\delta + \lambda(1 - F(V(s|w(.))))]ds}
    \end{align}
  \item Define $G(V)$ the steady state number of workers with lifetime utility less than $V$.
  \item Then,
    \begin{align}
      \Omega &= \underbrace{[\lambda G(V_{0})]}_{Finding\;Rate}\int_{0}^{\infty}\underbrace{\psi(t|w(.))}_{Survival}\underbrace{[p - w(t)]}_{Flow\;profits}dt
    \end{align}
  \item Where $\Omega$ is the equilibrium profits of a firm posting contract $w(.)$ s.t., $E[u(w(.))] = V_{0}$.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Optimal Contract}
  \begin{itemize}
  \item Fundamental question: Why do we write contracts?
  \item Firm dynamic programming problem
    \begin{align}
      \max_{w(.)\geq 0}\int_{0}^{\infty}\psi(t|w(.))[p - w(t)]dt
    \end{align}
  \item What are the two terms here?
    \begin{align}
      \max_{w(.)\geq 0}\int_{0}^{\infty}(\underbrace{\psi(t|w(.))p}_{Gains} - \underbrace{\psi(t|w(.))w(t)}_{Losses})dt
    \end{align}
  \item Subject to $V(0|w(.)) = V_{0}$.
  \item ``Free Entry'':
    \begin{align}
      \Omega^{*}(V_{0}) = \lambda G(V_{0})\pi^{*}(0|V_{0})
    \end{align}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Equilibrium}
  \begin{itemize}
  \item Assumptions:
    \begin{enumerate}
    \item Convex preferences: $u, > 0$, $u'' < 0$ and exists, and $\lim_{c\leftarrow 0}u(c) = -\infty$
    \item $\forall V_{0}\in(\underline{V},\bar{V})$, $F$ is continuously diff'ble \& $F'(V_{0}) > 0$.
    \end{enumerate}
  \item From the paper:\\
{\it A market equilibrium} is:
\begin{enumerate}
\item a distribution of starting payoffs $F$.
\item Optimal wage-tenure contracts $w^{*}(.|V_{0})$, $V_{0}\geq V_{U}$.
\item Optimal stopping-time solutions to both worker problems.
\item Worker value distribution $G$ consistent with worker flows.
\item Optimal posting game:
  \begin{align}
    \Omega^{*}(V_{0}) &= \bar{\Omega} \forall V_{0}\in[\underline{V},\bar{V}]\\
    \Omega^{*}(V_{0}) &\leq \bar{\Omega} \text{ otherwise}
  \end{align}
\end{enumerate}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Optimal Contract}
  \begin{itemize}
  \item Fundamental question: Why do we write contracts?
  \item Firm dynamic programming problem
    \begin{align}
      \max_{w(.)\geq 0}&\int_{0}^{\infty}\psi(t|w(.))[p - w(t)]dt\\
      \text{s.t.  } w(.)&\geq 0\\
      V(0|w(.)) &= V_{0}
    \end{align}
  \item Continuous-time optimization (Hamiltonian) where
    \begin{align}
      \dot{\psi} = -[\delta + \lambda(1 - F(V))]\psi\\
      \dot{V} = \delta V - u(w) - \lambda\int_{V}^{\bar{V}}[x - V]dF(x)
    \end{align}
  \item Hamiltonian:
    \begin{align}
      H &= \psi[p - w] + x_{\psi}\dot{\psi} + x_{V}\dot{V}
    \end{align}
  \item Optimal path of the wage would solve this problem.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Optimal Contract}
  \begin{itemize}
  \item Solving the contracting problem yields the following contract:
    \begin{align}
      \frac{u'(w^{*}(0|.))}{u'(w^{*}(\tau|.))} = 1 + u'(w^{*}(0|.))\int_{0}^{\tau}\lambda F'(V^{*}(t|.))\pi^{*}(t|.)dt
    \end{align}
  \item Strictly increasing and concave utility:
    \begin{align}
      \frac{u'(w^{*}(0|.))}{u'(w^{*}(\tau|.))} > 1 \rightarrow w^{*}(\tau) > w^{*}(0)
    \end{align}
  \item Increasing wage at time $\tau$ decreases quit rate over $[0, \tau]$:
    \begin{align}
      \underbrace{u'(w^{*}(0|.))}_{Slope}\int_{0}^{\tau}\underbrace{\lambda F'(V^{*}(t|.))}_{\Delta\;Quit\;Rate}\underbrace{\pi^{*}(t|.)}_{PDV\;of\;Profits}dt
    \end{align}
  \item Moral hazard: reward those who do not quit.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Market Equilibrium}
  \begin{itemize}
  \item Similar to the BM model (but with more algebra), can show that there exists a unique equilibrium, with the following:
    \begin{align}
      (\frac{\delta}{\lambda + \delta})^{2} &= \frac{p - \bar{w}}{p - \underline{w}}\\
      u(\underline{w}) &= u(b) - \frac{\sqrt{p - \underline{w}}}{2}\int_{\underline{w}}^{\bar{w}}\frac{u'(x)dx}{\sqrt{p - x}}
    \end{align}
  \item With a wage-tenure contract given by
    \begin{align}
      \frac{dw}{dt} = \frac{\delta}{\sqrt{p - \bar{w}}}\frac{p - w}{u'(w)}\int_{w}^{\bar{w}}\frac{u'(x)dx}{\sqrt{p - x}}
    \end{align}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Market Equilibrium}
  \begin{itemize}
  \item Pins down initial wage \& thus contract. (Burdett and Coles, 2003)
    \centering\includegraphics[width=0.9\textwidth]{BurdettColes1.png}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Risk Aversion}
  \begin{itemize}
  \item We breezed through preferences earlier.
  \item What role does consumption risk play here?
    \begin{enumerate}
    \item Way to give workers a preference over when they receive consumption.
    \item Opportunity for firm to provide insurance against shocks.
    \end{enumerate}
  \item They use different CRRA thresholds: (Burdett and Coles, 2003)\\
    \centering\includegraphics[width=0.9\textwidth]{BurdettColes2.png}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Consumption Risk}
  \begin{itemize}
  \item What is strange about this table? (Burdett and Coles, 2003)
\begin{center}
\includegraphics[width=0.8\textwidth]{BurdettColes2.png}
\end{center}
  \item There was no precautionary savings here.
  \item i.e., firms attracted workers by offering {\it insurance} against low consumption via a guaranteed contract.
  \item But, transitory consumption risk already mitigated by autarky and UI.
  \end{itemize}
\end{frame}

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

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

\begin{frame}
  \frametitle{Next Time}
  \begin{itemize}
  \item Postel-Vinay and Robin (2002): Sequential auctions (i.e., firm {\it can} respond to outside offers)
  \item Next Thursday, introduction/research proposal due.
  \item Presentations the following Tuesday (probably).
  \item Start your data projects soon!
  \end{itemize}
\end{frame}


\end{document}