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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 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{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 There is a new homework online.
  \item All should have access to the cluster.
  \item Today:
    \begin{itemize}
    \item Talk about Lucas Critique and Rational Expectations
    \item Use simple two period model
    \item Show intuition behind Lucas Critique.
    \end{itemize}
  \item Lecture largely based on Eric Sims' (Notre Dame) notes.
  \end{itemize}
\end{frame}

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

\section{A Two-Period 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{Lucas Critique Overview}
  \begin{itemize}
  \item Some history:
    \begin{itemize}
    \item Prior to the late 1970s, macroeconomists had no systematic way of modeling consumer expectations.
    \item They found \textit{empirical relationships} between \textit{equilibrium objects} and interpreted these as causal.
    \item This is a problem!
    \item (Old) Phillips Curve: inverse relationship between inflation and unemployment
      \begin{itemize}
      \item more money $\rightarrow$ more demand $\rightarrow$ more employment.
      \end{itemize}
    \item This led policy makers to institute persistent inflation.
    \item But this broke down in the 70s: we had stagflation: inflation and unemployment.
    \item The reason is that consumers came to expect an increase in prices and adjusted their demand.
    \end{itemize}
  \item Lucas Critique (broadly):
    \begin{itemize}
    \item Need to use ``deep'' (structural) parameters to inform policy.
    \item Otherwise policy may affect these parameters.
    \end{itemize}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Basic two-period model}
  \begin{itemize}
  \item A (very) basic consumption-savings model:
    \begin{align}
      \max_{c_{1},c_{2}} u(c_{1}) &+ \beta u(c_{2})\\
      \text{s.t.  } c_{1} + \frac{c_{2}}{1 + r} &= w_{1} + T_{1} + \frac{w_{2}}{1 + r} + \frac{T_{2}}{1 + r}
    \end{align}
  \item Simple set-up:
    \begin{itemize}
    \item Household receives permanent income $(w_{1}, w_{2})$, which might have a dependence structure.
    \item They also may receive transitory income $(T_{1}, T_{2})$, which does not have a dependence structure.
    \item Define $y_{t} = w_{t} + T_{t}$
    \item $r$ is fixed over time, household takes as given.
    \item Standard definitions for $u$: $u' > 0$, $u'' < 0$, $u'(0) = \infty$
    \end{itemize}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Basic two-period model}
  \begin{itemize}
  \item A (very) basic consumption-savings model:
    \begin{align}
      V = \max_{c_{1},c_{2}} u(c_{1}) &+ \beta u(c_{2})\\
      \text{s.t.  } c_{1} + \frac{c_{2}}{1 + r} &= y_{1} + \frac{y_{2}}{1 + r}
    \end{align}
  \item Solve by first finding the Euler Equation:
    \begin{align}
      \der{V}{c_{1}} &= u'(c_{1}) - \lambda = 0\\
      \der{V}{c_{2}} &= \beta u'(c_{2}) - \frac{\lambda}{1 + r} = 0\\
      \rightarrow u'(c_{1}) &= \beta(1 + r)u'(c_{2})
    \end{align}
  \item We know dynamics, now need to pin down $c_{t}$ using budget constraint (boundary condition).
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Basic two-period model}
  \begin{itemize}
  \item Dynamics and budget:
    \begin{align}
      u'(c_{1}) &= \beta(1 + r)u'(c_{2})\\
      \text{s.t.  } c_{1} + \frac{c_{2}}{1 + r} &= y_{1} + \frac{y_{2}}{1 + r}
    \end{align}
  \item Assume log utility: $u(c) = ln(c)$.
  \item This yields
    \begin{align}
      \frac{1}{c_{1}} &= \beta(1 + r)\frac{1}{c_{2}}\\
      c_{1} &= \frac{1}{1 + \beta}(y_{1} + \frac{y_{2}}{1 + r})
    \end{align}
  \item What does this tell us?
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Basic two-period model}
    \begin{align}
      c_{1} &= \frac{1}{1 + \beta}(y_{1} + \frac{y_{2}}{1 + r})
    \end{align}
  \begin{itemize}
  \item This tells us that consumption today is a function of
    \begin{itemize}
    \item income today (not surprising)
    \item income in the future (possibly a problem)
    \end{itemize}
  \item Suppose there is a recession.
  \item A policymaker wants to implement a tax cut based on empirical evidence
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Basic two-period model}
\begin{itemize}
  \item (Assume $\beta = \frac{1}{1 + r}$ so that $c_{2} = c_{1} = c_{t}$.
    \begin{align}
      c_{t} &= \frac{1}{1 + \beta}(y_{t} + \frac{y_{t+1}}{1 + r})
    \end{align}
  \item Policymaker:
    \begin{itemize}
    \item Run the following regression:
      \begin{align}
        c_{t} = \alpha + \gamma y_{t} + u_{t}
      \end{align}
    \item Want to stimulate the economy.
    \item Give people money, consumption will increase by $\gamma$!
    \end{itemize}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Basic two-period model}
    \begin{align}
      c_{t} &= \frac{1}{1 + \beta}(w_{t} + T_{t} + \frac{w_{t + 1}}{1 + r} + \frac{T_{t + 1}}{1 + r})\\
      \hat{c_{t}} &= \alpha + \gamma y_{t}
    \end{align}
  \begin{itemize}
  \item Policy maker changes transitory income, $T_{t}$. Problem?
  \item Assume $\der{w_{2}}{w_{1}} = 0$ (i.e., uncorrelated). Then
    \begin{align}
      \der{c_{t}}{w_{t}} &= \der{c_{t}}{T_{t}} = \frac{1}{1 + \beta}\\
      \der{\hat{c}_{t}}{w_{t}} &= \der{\hat{c}_{t}}{T_{t}} =\gamma
    \end{align}
  \item In this context, $\gamma = \frac{1}{1 + \beta}$. We're good!
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Basic two-period model}
    \begin{align}
      c_{t} &= \frac{1}{1 + \beta}(w_{t} + T_{t} + \frac{w_{t + 1}}{1 + r} + \frac{T_{t + 1}}{1 + r})\\
      \hat{c_{t}} &= \alpha + \gamma y_{t}
    \end{align}
  \begin{itemize}
  \item Assume $\der{w_{2}}{w_{1}} \cancel{=} 0$. Then
    \begin{align}
      \der{c_{t}}{y_{t}} &= \frac{1}{1 + \beta}(1 + \frac{\der{w_{t+1}}{w_{t}}\der{w_{t}}{y_{t}}}{1 + r})\\
      \der{\hat{c}_{t}}{y_{t}} &= \gamma > \der{c_{t}}{T_{t}}\\
      \der{c_{t}}{T_{t}} &= \frac{1}{1 + \beta}
    \end{align}
  \item If income was positively correlated (AR, etc.), we're not going to get the response we want.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Lucas critique overview}
  \begin{itemize}
  \item In this context, policymakers might over predict the response of consumption.
  \item Why? Because consumers understand that this is a temporary increase in income.
  \item They won't believe that $w_{2}$ will increase.
  \item Therefore, they will respond less than predicted by the model.
  \item This is the crux of the Lucas Critique: that you need to find deep parameters that don't change with consumer behavior.
  \end{itemize}
\end{frame}

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

\section{Monetary policy}
% ------------------------------------------------

\begin{frame}
  \frametitle{Lucas critique}
  \begin{itemize}
  \item Lucas made his critique in the context of monetary policy.
  \item There had been multiple decades of inflation, aimed at reducing unemployment.
  \item Consumers eventually built in the expectation of inflation and this empirical relationship no longer held.
  \item Let's use a simple monetary policy model to understand what happened.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Phillips Curve}
  \begin{itemize}
  \item Suppose that inflation is characterized by the following difference equation:
    \begin{align}
      \pi_{t} = \theta(u_{t} - u^{*}) + \beta\mathbb{E}(\pi_{t + 1})
    \end{align}
  \item What does this tell us?
    \begin{itemize}
    \item If we hold expectations fixed,
    \item an increase in current inflation, $\pi_{t}$,
    \item leads to a $\theta$ reduction in unemployment (in percentage points).
    \end{itemize}
  \item $\theta$ was observed to be negative, ie inflation reduced unemployment.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Policymaker}
    \begin{align}
      \pi_{t} = \theta(u_{t} - u^{*}) + \beta\mathbb{E}(\pi_{t + 1})
    \end{align}
  \begin{itemize}
  \item Suppose that an econometrician ran the following specification:
    \begin{align}
      \pi_{t} = \gamma (u_{t} - u^{*}) + \epsilon_{t}
    \end{align}
  \item They conclude that $\gamma < 0$.
  \item They tell the policymaker to raise inflation to reduce unemployment.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{What happens?}
    \begin{align}
      \pi_{t} &= \theta(u_{t} - u^{*}) + \beta\mathbb{E}(\pi_{t + 1})\\
      \pi_{t} &= \gamma (u_{t} - u^{*}) + \epsilon_{t}
    \end{align}
  \begin{itemize}
  \item Well, as long as expectations don't change, the empirical specification will appear to hold.
  \item But if they change, consequences!
  \item An increase in inflation can lead to one of two things:
    \begin{enumerate}
    \item a decrease in unemployment (good!) or
    \item an increase in expected future inflation
    \end{enumerate}
  \item and the equation will still hold.
  \item This is what we saw in the 1970s/1980s.
  \end{itemize}
\end{frame}

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

\section{Log linearization} % 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{Log linearization}
  \begin{itemize}
  \item Non-linear difference equations are tricky to solve.
  \item Macroeconomists often log-linearize these difference equations to make them easier to solve.
  \item Basic idea:
    \begin{itemize}
    \item In some area around the steady state, deviations are small.
    \item Can approximate using a log-linearized version of the model.
    \item Will be ``wrong,'' but close as long as deviations stay small.
    \end{itemize}
  \item Today, short refresher.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Log linearization II}
  \begin{itemize}
  \item Take a generic difference equation with a single variable $x$:
    \begin{align}
      x_{t + 1} = A x_{t}
    \end{align}
  \item Suppose that $A = 1 + g$:
    \begin{align}
      x_{t + 1} = (1 + g)x_{t}
    \end{align}
  \item Taking logs of both sides:
    \begin{align}
      ln(x_{t + 1}) = ln(1 + g) + ln(x_{t})
    \end{align}
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Taylor Series Approximation}
  \begin{itemize}
  \item A first-order taylor approximation of a function $f(x)$ around a point $x^{*}$ is given by
    \begin{align}
      f(x) \approx f(x^{*}) + f'(x^{*})(x - x^{*})
    \end{align}
  \item If $(x - x^{*})$ is small and $f''$ is not too large, this approximation is reasonable.
  \item Idea: we know the value of a function at a particular point
  \item We can also find the derivative at that point.
  \end{itemize}
\end{frame}

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

\begin{frame}
  \frametitle{Applying this to log-linear approximation}
  \begin{itemize}
  \item Taylor series approximation of growth rate $(1 + g)$ at $g = 0$:
    \begin{align}
      ln(1 + g) &\approx ln(1 + 0) + \frac{1}{1 + 0}(1 + g - 1)\\
                &\approx g
    \end{align}
  \item This means that we can approximate our difference equation as
    \begin{align}
      ln(x_{t + 1}) &= ln(1 + g) + ln(x_{t})\\
                    &\approx ln(x_{t}) + g
    \end{align}
  \item This insight will prove very useful in solving macro models.
  \end{itemize}
\end{frame}

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

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


\begin{frame}
  \frametitle{Next Time}
  \begin{itemize}
  \item Start dynamic programming.
  \item Homework due 2/19 (next Thursday).
  \end{itemize}
\end{frame}


\end{document}