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% ----------------------------------------------------------------------------------------
%	TITLE PAGE
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\title[]{Quantitative Macro-Labor:\\ Solving through Linearizing} % 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 2022} % 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 Today: Start discussing solution techniques.
  \item Focus on linearization.
  \item Discuss the upsides and downsides.
  \item Next time: global techniques.
  \item Empirical regularities project is due in two weeks!
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Solving a Model}
  \begin{itemize}
  \item When we say ``solve a model'' what do we mean?
    \begin{enumerate}
    \item Find the equilibrium of the model.
    \item Generally, determine the policy functions.
    \item Determine the transition equations given the individual and aggregate state.
    \item i.e., aggregate up the policy functions and determine prices given distributions.
    \end{enumerate}
  \item Generically, this is hard: many states, non-linear decision rules, etc.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Solving a Model}
  \begin{itemize}
  \item Generically, this is hard: many states, non-linear decision rules, etc.
  \item Much of quantitative macro is about finding ``shortcuts'' without sacrificing accuracy of solution (some we have seen):
    \begin{enumerate}
    \item Planner's problem: use welfare theorems to remove prices from problem.
    \item Rational expectations \& complete markets: Aggregate worker decision rules by assuming they make same predictions about future prices, and face same consumption risk.
    \item Exogenous wage distribution/prices: agents do not respond to decisions of other agents.
    \item Block Recursive Equilibrium: agents face an equilibrium with individual prices, i.e., no need to know distribution.
    \end{enumerate}
  \item Linearization: assume the economy is close enough to steady-state that transition equations (i.e., policy functions) are close to linear within small deviations.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Linearization - Intuition}
  \begin{itemize}
  \item Approximate a function around point where second derivative small (in abs. terms)
  \item Our models: around steady-state.
  \end{itemize}
\centering\includegraphics[width=0.8\textwidth]{Linearization1.png}
\end{frame}

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

\begin{frame}
  \frametitle{Linearization - Basic Approach}
  \begin{itemize}
  \item Write model to incorporate out of steady-state equilibria in expectations.
  \item If not: just comparative statics.
  \item Select point around which changes in dynamics are small.
  \item Use Taylor Series approximation evaluated at point.
  \item Taylor Series:
    \begin{align}
      f(x) \approx f(a) + \frac{f'(a)}{1!}(x - a) + \frac{f''(a)}{2!}(x - a)^{2} + ...
    \end{align}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Linearization - Example}
  \begin{itemize}
  \item Taylor Series:
    \begin{align}
      f(x) \approx f(a) + \frac{f'(a)}{1!}(x - a) + \frac{f''(a)}{2!}(x - a)^{2} + ...
    \end{align}
  \item Ex: approximate function $f(x) = ln(1 + r)$ for small $r$.
    \begin{align}
      f(x) &= ln(1 + r)\\
           &\approx ln(1) + \frac{1}{1}((1 + r) - 1)\\
           &\approx r\\
    \end{align}
  \item This is why we coefficients in regressions are interpreted as percent change when LHS is in logs.
  \end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{Local Linear Methods}
\begin{itemize}
\item Log-linearize the system around the steady-state, then proceed.
\item In general, must determine linearized equilibrium, then apply solution algorithm.
\item Solution algorithms:
\begin{enumerate}
\item Klein's Method (2000): Used for singular matrices.
\item Sim's Method (2001): Used when it is unclear which variables are states and controls.
\item Blanchard and Kahn's Method (1980): First solution method for rational expectations models.
\end{enumerate}
\item Basic idea of these techniques: find $x(0)$ such that system arrives at steady-state.
\end{itemize}
\end{frame}

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

\section{The Mortensen and Pissarides Model}
% ------------------------------------------------

\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 $y = z>b$.
  \item Matching function:
    \begin{enumerate}
    \item Determines {\it number} of meetings between firms \& workers.
    \item Args: levels searchers \& vacancies $(U = u\times L, V = v\times L$)
    \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}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Value Functions}
  \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}
      rV &= 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}

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

\begin{frame}
  \frametitle{Equilibrium Objects}
  \begin{itemize}
  \item Three key equilibrium objects:
    \begin{enumerate}
    \item Wages;
    \item unemployment;
    \item $\theta = \frac{v}{u}$ (vacancies).
    \end{enumerate}
  \item How do we solve for each if we can't assume the model is in the steady-state?
  \item i.e., what happens when model economy is in {\it transition}?
  \item Policy functions ($w,\theta$) are forward-looking: must re-solve model to incorporate dynamics.
  \item Intuition: high or low productivity periods may change asset value of jobs.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Worker Value Functions}
  \begin{itemize}
  \item Value functions:
    \begin{enumerate}
    \item Employed at wage w: $W(w)$
    \item Unemployed: $U$.
    \end{enumerate}
  \item Unemployed flow value:
    \begin{align}
      rU = b + p(\theta)[W(w) - U] + \dot{U}
    \end{align}
  \item Employed flow value:
    \begin{align}
      rW(w) = w + \delta[U - W(w)] + \dot{W(w)}
    \end{align}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Firm Value Functions}
  \begin{itemize}
  \item Value functions:
    \begin{enumerate}
    \item Filled, paying wage w: $J(w)$
    \item Vacant $V$.
    \end{enumerate}
  \item Vacant flow value:
    \begin{align}
      rV = -\kappa + q(\theta)[J(w) - V] + \dot{V}
    \end{align}
  \item Matched flow value:
    \begin{align}
      rJ(w) = (z - w) + \delta[V - J(w)] + \dot{J(w)}
    \end{align}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Surplus}
  \begin{itemize}
  \item I emphasized in endogenous separation lecture, but surplus again key.
  \item Search and matching models with Nash Bargaining: surplus determines all policy functions.
  \item Surplus:
    \begin{align}
      S(w) &= W(w) + J(w) - U - V\\
      \dot{S(w)} &= \dot{W(w)} + \dot{J(w)} - \dot{U} - \dot{V}
    \end{align}
  \item Similarly, dynamics of surplus equation will determine all dynamics of model.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Assumptions}
  \begin{itemize}
  \item To solve this model, we will make two assumptions about dynamics:
    \begin{enumerate}
    \item Free entry {\bf always} holds;
    \item Wages are continuously renegotiated.
    \end{enumerate}
  \item Meaning:
    \begin{align}
      V &= 0, \dot{V} = 0\\
      w &= \beta S(w)\\
      \dot{w} &= \beta \dot{S}(w)
    \end{align}
  \item If these were only two equilibrium objects: aggregate shocks would immediately move model to new steady-state.
  \item But unemployment is {\bf not} a jump variable.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Dynamics}
  \begin{itemize}
  \item Know from free entry that
    \begin{align}
      q(\theta) &= \frac{\kappa}{J(w)}\\
      J &= (1 - \beta)S\\
      \rightarrow S &= \frac{\kappa}{(1 - \beta)q(\theta)}\\
      \dot{S} &= -\frac{\kappa q'(\theta)\dot{\theta}}{(1 - \beta)(q(\theta))^{2}}
    \end{align}
  \item Surplus:
    \begin{align}
      (r + \delta)S &= z - b + p(\theta)\beta S + \dot{S}\\
      \rightarrow 0 &= \frac{\kappa q'(\theta)\dot{\theta}}{(q(\theta))^{2}} + \frac{\kappa(r + \delta + p(\theta)\beta)}{q(\theta)} - (1 - \beta)(z - b)
    \end{align}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Dynamics II}
  \begin{itemize}
  \item Wages continuously renegotiated, thus change with surplus.
  \item Only equilibrium object that does not jump: unemployment.
  \item Recall flow equation for unemployment:
    \begin{align}
      \dot{u} &= \delta(1 - u) - p(\theta)u
    \end{align}
  \item By free entry condition, we know that $\theta = \theta^{*}$. Vacancies adjust to new steady-state ratio.
    \begin{align}
      \rightarrow \dot{u} &= \delta(1 - u) - p(\theta^{*})u
    \end{align}
  \item $\dot{u}\cancel{=}0$? Because firms may hire more or less of unemployed pool.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Linearization}
  \begin{itemize}
  \item Thus far, we haven't taken a stand on our solution technique.
  \item We have equations that determine out of steady-state dynamics for
    \begin{enumerate}
    \item Surplus: $\theta$ and wages;
    \item Unemployment.
    \end{enumerate}
  \item Rest of class: use this as jumping off point to solve linearized version of model.
  \item What do we need to linearize? Surplus equation.
  \item Approach:
    \begin{enumerate}
    \item Compute steady-state.
    \item Linearize around steady-state.
    \item Solve linear system.
    \end{enumerate}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Linearization II}
  \begin{itemize}
  \item Steady-state $\theta$:
    \begin{align}
      0 &= \frac{\kappa q'(\theta)\dot{\theta}}{(q(\theta))^{2}} + \frac{\kappa(r + \delta + p(\theta)\beta)}{q(\theta)} - (1 - \beta)(z - b)\\
      &\rightarrow (1 - \beta)(z - b) = \frac{\kappa(r + \delta + \theta q(\theta^{*})\beta)}{q(\theta^{*})}
    \end{align}
  \item From here:
  \item Solve for $\dot{\theta} = f(\theta)$.
  \item Linearize:
    \begin{align}
      f(\theta) \approx f(\theta^{*}) + f'(\theta^{*})(\theta - \theta^{*})
    \end{align}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Linearization III}
  \begin{itemize}
  \item Steady-state $\theta$:
    \begin{align}
      0 &= \frac{\kappa q'(\theta)\dot{\theta}}{(q(\theta))^{2}} + \frac{\kappa(r + \delta + p(\theta)\beta)}{q(\theta)} - (1 - \beta)(z - b)
    \end{align}
  \item Solving for $\dot{\theta} = f(\theta)$:
    \begin{align}
      \dot{\theta} &= \frac{\beta q(\theta)^{2}}{q'(\theta)}(\frac{1 - \beta}{\beta}\frac{z - b}{\kappa} - \frac{r + \delta}{\beta q(\theta)} - \theta)
    \end{align}
  \item Taylor approximation of $\dot{\theta}$ around $\theta^{*}$:
    \begin{align}
      \dot{\theta} &\approx \cancel{f(\theta^{*})} + [(r + \delta) - \frac{\beta q(\theta^{*})^{2}}{q'(\theta^{*})}](\theta - \theta^{*})\\
      \dot{\theta} &\approx A\Delta\theta
    \end{align}
  \item What does $A > 0$ mean?
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Stability}
  \begin{itemize}
  \item Taylor approximation of $\dot{\theta}$ around $\theta^{*}$:
    \begin{align}
      \dot{\theta} &\approx \cancel{f(\theta^{*})} + [(r + \delta) - \frac{\beta q(\theta^{*})^{2}}{q'(\theta^{*})}](\theta - \theta^{*})\\
      \dot{\theta} &\approx A\Delta\theta
    \end{align}
  \item If $A > 0$, no solution.
  \item Here: must be that $\dot{\theta} = 0$.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Unemployment}
  \begin{itemize}
  \item Only dynamics: unemployment.
    \begin{align}
      \dot{u} &= \delta(1 - u) - p(\theta^{*})u
    \end{align}
  \item If $\theta_{2}^{*} << \theta_{1}^{*}$, $u\uparrow$.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Bad Example of Linearizing}
  \begin{itemize}
  \item Ex-post: this wasn't the best model to explore linearization methods in.
  \item Euler Equation from Hansen (1985):
    \begin{align}
      % y_{t} &= a_{t}k_{t}^{\theta}h^{1 - \theta}\\
      % ln(a_{t}) &+ (1 - \rho)ln(A) + \rho ln(a _{t - 1}) + \epsilon_{t}\\
      % y_{t} &= c_{t} + i_{t}\\
      % \eta k_{t + 1} &= (1 - \delta)k_{t} + i_{t}\\
      % \gamma c_{t}h_{t} &= (1 - \theta)y_{t}\\
      \frac{\eta}{c_{t}} &= \beta E_{t}[\frac{1}{c_{t + 1}}(\theta\frac{y_{t + 1}}{k_{t + 1}} + 1 - \delta)]
    \end{align}
  \item Next steps: solve for steady-state.
  \item Here: show how to linearize this model
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Linearized Euler Equation}
  \begin{itemize}
  \item Define deviations from steady-state: $\tilde{x}_{t} = ln(x_{t}) - ln(x)$:
    \begin{align*}
      \tilde{x}_{t} &\approx \tilde{x}_{t}(x) + \der{\tilde{x}_{t}}{x_{t}}(x)(x_{t} - x)\\
      &\approx ln(1) + \frac{1}{x}(x_{t} - x)\\
      &\approx x(1 + \tilde{x}_{t})
    \end{align*}
    \begin{align*}
      \frac{\eta}{c_{t}} &= \beta E_{t}[\frac{1}{c_{t + 1}}(\theta\frac{y_{t + 1}}{k_{t + 1}} + 1 - \delta)]\\
      \frac{\eta}{ce^{\tilde{c}_{t}}} &= \beta E_{t}[\frac{1}{ce^{\tilde{c}_{t + 1}}}(\theta(\frac{ye^{\tilde{y}_{t + 1}}}{ke^{\tilde{k}_{t + 1}}}) + 1 - \delta)]\\
      \frac{\eta}{\beta}e^{-\tilde{c}_{t}} &= E[e^{-\tilde{c}_{t + 1}}(\theta\frac{y}{k}e^{\tilde{y}_{t + 1}}e^{-\tilde{k}_{t + 1}} + 1 - \delta)]\\
      \rightarrow\frac{\eta}{\beta}(1 - \tilde{c}_{t}) &= E[(1 - \tilde{c}_{t + 1})(\theta\frac{y}{k}(1 + \tilde{y}_{t + 1})(1 - \tilde{k}_{t + 1}) + 1 - \delta)]
    \end{align*}
  \item Using $e^{-\tilde{x}_{t}}\approxeq (1 - \tilde{x}_{t})$.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Linearized Euler Equation II}
  \begin{itemize}
  \item Percent change X percent change $\approx 0$ ($\tilde{x}_{t}\times\tilde{y}_{t}\approx 0$)
    \begin{align}
      \frac{\eta}{\beta}(1 - \tilde{c}_{t}) &= E[(1 - \tilde{c}_{t + 1})(\theta\frac{y}{k}(1 + \tilde{y}_{t + 1} - \tilde{k}_{t + 1}) + 1 - \delta)]\nonumber\\
      \frac{\eta}{\beta}(1 - \tilde{c}_{t}) &= E[\theta\frac{y}{k}(1 + \tilde{y}_{t + 1} - \tilde{k}_{t + 1}) + 1 - \delta \nonumber\\&- \tilde{c}_{t + 1}\theta\frac{y}{k} - \tilde{c}_{t + 1} + \delta\tilde{c}_{t + 1}]
    \end{align}
  \item Steady-state value of capital given by
    \begin{align}
      k^{*} = (\frac{\theta}{\frac{\eta}{\beta} - 1 + \delta})y^{*}
    \end{align}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Linearized Euler Equation III}
  \begin{itemize}
  \item Percent change X percent change $\approx 0$ ($\tilde{x}_{t}\times\tilde{y}_{t}\approx 0$)
    \begin{align}
\frac{\eta}{\beta}(1 - \tilde{c}_{t}) &= E[\theta\frac{y}{(\frac{\theta}{\frac{\eta}{\beta} - 1 + \delta})y}(1 + \tilde{y}_{t + 1} - \tilde{k}_{t + 1}) \nonumber\\&+ 1 - \delta - \tilde{c}_{t + 1}\theta\frac{y}{(\frac{\theta}{\frac{\eta}{\beta} - 1 + \delta})y} - \tilde{c}_{t + 1} + \delta\tilde{c}_{t + 1}]\\
      \frac{\eta}{\beta}(1 - \tilde{c}_{t}) &= E[(\frac{\eta}{\beta} - 1 + \delta)(1 + \tilde{y}_{t + 1} - \tilde{k}_{t + 1}) \nonumber\\&+ 1 - \delta - (\frac{\eta}{\beta} - 1 + \delta)\tilde{c}_{t + 1} - \tilde{c}_{t + 1} + \delta\tilde{c}_{t + 1}]
    \end{align}
  \item Yielding the following linearized Euler Equation:
    \begin{align}
      0 = \frac{\eta}{\beta}\tilde{c}_{t} + E[(\frac{\eta}{\beta} - 1 + \delta)(\tilde{y}_{t + 1} - \tilde{k}_{t + 1}) - \frac{\eta}{\beta}\tilde{c}_{t + 1}]
    \end{align}
  \item This is precisely the form we want: $0 = f(x)$.
  \end{itemize}
\end{frame}

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

\begin{frame}
\frametitle{Log-Linearizing the System}
\begin{itemize}
\item Static variables: output ($y_{t}$), hours ($h_{t}$), investment ($i_{t}$).
\item Dynamic variables: capital ($k_{t}$) and consumption ($c_{t}$).
\item Stochastic dynamic variable: productivity ($a_{t}$).
\item We can now write the system as:
\begin{equation}
\Psi_{1}\zeta_{t} = \Psi_{2}\xi_{t} + \Psi_{3}\tilde{a}_{t}
\end{equation}
\begin{equation}
\Psi_{4}E_{t}(\xi_{t + 1}) = \Psi_{5}\xi_{t} + \Psi_{6}\zeta_{t} + \Psi_{7}\tilde{a}_{t}
\end{equation}
\item $\zeta_{t}$ are static predetermined and nonpredetermined variables, $[\tilde{y}_{t}, \tilde{h}_{t}, \tilde{i}_{t}]'$.
\item $\xi_{t}$ are dynamic predetermined and nonpredetermined variables, $[\tilde{k}_{t}, \tilde{c}_{t}]'$.
\item $\tilde{a}_{t}$ is the technology process.
\item Why is $\tilde{c}_{t}$ among the dynamic variables?
\end{itemize}
\end{frame}

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

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

\begin{frame}
  \frametitle{Next Time}
  \begin{itemize}
  \item Global Solutions to the DMP model.
  \item Empirical regularities project due very soon!
  \end{itemize}
\end{frame}


\end{document}
