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% ----------------------------------------------------------------------------------------
%	TITLE PAGE
% ----------------------------------------------------------------------------------------

\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 2021} % 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: Continue the New Keynsian Model
\item RBC model: nominal variables are neutral.
\item New Keynesian model:
\begin{enumerate}
\item Nominal rigidities: prices are sticky, i.e. slow to adjust.
\item This leads to a role for stabilization policies.
\end{enumerate}
\item Here: Linearize and show uniqueness of equilibrium
\end{itemize}
\end{frame}

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

\section{General Assumptions}

\begin{frame}\frametitle{Assumptions}
\begin{itemize}
\item Simplifying assumption:
  \begin{itemize}
  \item ignore variation in capital or investment.
  \end{itemize}

\item Prices are determined by ``Calvo (1983) Pricing'':
\begin{itemize}
\item prices are allowed to change with fixed probability
\end{itemize}
\item Wages are not sticky

\item Monetary policy is a choice of the nominal interest rate
\end{itemize}
\end{frame}

\section{Uniqueness of the equilibrium}

\begin{frame}\frametitle{Basic Model}
\begin{itemize}
\item New Keynesian Phillips Curve
\[
\pi _{t}=\kappa x_{t}+\beta E_{t}\pi _{t+1}
\]
\item Output gap $x_{t}=\hat{y}_{t}-\hat{y}%
_{t}^{f} $%
\[
x_{t}=E_{t}x_{t+1}-\frac{1}{\sigma }\left( \hat{\imath}_{t}-E_{t}\pi
_{t+1}\right) +u_{t}
\]%
where $u_{t}=E_{t}\hat{y}_{t+1}^{f}-\hat{y}_{t}^{f}$

\item Taylor Rule for nominal interest rate%
\[
\hat{\imath}_{t}=\delta _{\pi }\pi _{t}+\delta _{x}x_{t}+v_{t}
\]
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Matrix form}
\begin{itemize}
\item Substitute for the interest rate to get a two-equation system in two
unknowns%
\[
E_{t}\pi _{t+1}=\frac{1}{\beta }\left[ \pi _{t}-\kappa x_{t}\right]
\]%
\[
E_{t}x_{t+1}=x_{t}+\frac{1}{\sigma }\left[ \delta _{\pi }\pi _{t}+\delta
_{x}x_{t}+v_{t}\right] -u_{t}-\frac{1}{\sigma \beta }\left[ \pi _{t}-\kappa
x_{t}\right]
\]

\item In matrix form%
\[
\left[
\begin{array}{c}
E_{t}\pi _{t+1} \\
E_{t}x_{t+1}%
\end{array}%
\right] =\left[
\begin{array}{c}
\frac{1}{\beta }\hspace{0.3in}\hspace{0.3in}\frac{-\kappa }{\beta } \\
\frac{\beta \delta _{\pi }-1}{\sigma \beta }\hspace{0.3in}1+\frac{\beta
\delta _{x}+\kappa }{\beta \sigma }%
\end{array}%
\right] \left[
\begin{array}{c}
\pi _{t} \\
x_{t}%
\end{array}%
\right] +\left[
\begin{array}{c}
0 \\
\frac{v_{t}}{\sigma }-u_{t}%
\end{array}%
\right]
\]
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Uniqueness of equilibrium}
\begin{itemize}
\item For uniqueness of equilibrium, since there are two forward-looking
variables, the roots of the characteristic equation must both be greater
than one in absolute value

\begin{itemize}
\item equivalently, the system must be unstable.
\item Why? Single steady-state w/ unique equilibrium path.
\item More stable roots $\rightarrow$ equilibrium path not unique.
\end{itemize}

\item letting roots be $\theta $

\begin{itemize}
\item determinant is product of roots%
\begin{eqnarray*}
\theta _{1}\theta _{2} &=&\frac{1}{\beta }\left( 1+\frac{\beta \delta
_{x}+\kappa }{\beta \sigma }\right) +\frac{\kappa }{\beta }\left( \frac{%
\beta \delta _{\pi }-1}{\sigma \beta }\right) \\
&=&\frac{1}{\beta }\left( 1+\frac{\delta _{x}}{\sigma }+\frac{\kappa \delta
_{\pi }}{\sigma }\right)
\end{eqnarray*}
\item Stability requirement is
\[
\left( \theta _{1}-1\right) \left( \theta _{2}-1\right) >0
\]
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Some algebra later...}
\begin{itemize}
\item We get two conditions
\[
\delta _{x}\left( \frac{1}{\beta }-1\right) +\frac{\kappa }{\beta }\left(
\delta _{\pi }-1\right) >0
\]%
\[
\delta _{x}\left( 1-\beta \right) +\kappa \left( \delta _{\pi }-1\right) >0
\]%
\item Implying that the responses to the output gap and inflation must be large
enough
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Stability}
\begin{itemize}
\item Set expected changes to zero (strictly correct if shocks permanent)
\[
E_{t}\pi _{t+1}-\pi _{t}=\frac{1-\beta }{\beta }\pi _{t}-\frac{\kappa }{%
\beta }x_{t}=0
\]%
\[
E_{t}x_{t+1}-x_{t}=\frac{\beta \delta _{\pi }-1}{\sigma \beta }\pi _{t}+%
\frac{\beta \delta _{x}+\kappa }{\beta \sigma }x_{t}+\frac{v_{t}}{\sigma }%
-u_{t}=0
\]%
\[
\text{along }E_{t}\pi _{t+1}-\pi _{t}=0\hspace{0.3in}\text{ }\pi _{t}=\frac{%
\kappa }{1-\beta }x_{t}
\]

\item slope of $\Delta \pi =0$ is positive
\[
\text{along }E_{t}x_{t+1}-x_{t}=0\hspace{0.3in}\pi _{t}=\frac{\beta \delta
_{x}+\kappa }{1-\beta \delta _{\pi }}x_{t}+\frac{\beta }{1-\beta \delta
_{\pi }}\left( v_{t}-\sigma u_{t}\right)
\]

\item slope of $\Delta x=0$ depends on sign of $1-\beta \delta _{\pi }$%
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Case 1: $1-\beta \delta _{\pi }<0$}
\begin{itemize}
\item slope of $\Delta x=0$ is negative, implying that slope of $\Delta \pi
=0>$ slope of $\Delta x=0$
\[
\frac{\kappa }{1-\beta }>\frac{\beta \delta _{x}+\kappa }{1-\beta \delta
_{\pi }}
\]%
\[
\kappa \left( 1-\beta \delta _{\pi }\right) <\left( \beta \delta _{x}+\kappa
\right) \left( 1-\beta \right)
\]%
\[
-\kappa \beta \left( \delta _{\pi }-1\right) -\beta \delta _{x}\left(
1-\beta \right) <0
\]%
\begin{equation}
\kappa \left( \delta _{\pi }-1\right) +\delta _{x}\left( 1-\beta \right) >0
\label{ue}
\end{equation}

\item model is globally unstable, equivalently both roots are outside the
unit circle

\item unique equilibrium at intersection and otherwise explosion
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Cases 2 and 3}
\begin{itemize}
\item Case 2: $1-\beta \delta _{\pi }>0$ and slope of $\Delta \pi =0>$ slope
of $\Delta x=0$

\begin{itemize}
\item sign of (\ref{ue}) is reversed

\item model is saddlepath stable

\item can begin anywhere along the saddlepath and model approaches long-run
equilibrium
\end{itemize}

\item Case 3: $1-\beta \delta _{\pi }>0$ and slope of $\Delta \pi =0<$ slope
of $\Delta x=0$

\begin{itemize}
\item equation (\ref{ue}) holds (begin with sign reversed and do not flip on
second line)

\item model is globally unstable

\item unique equilibrium at intersection
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Necessary and sufficient conditions for uniqueness}
\begin{itemize}
\item Equation (\ref{ue}) is necessary and sufficient for global instability
and unique equilibrium

\begin{itemize}
\item uniqueness of equilibrium requires a strong enough response of the
interest rate to deviations of inflation and output from their steady-state
values
\end{itemize}
\end{itemize}
\end{frame}

\section{Monetary policy}

\begin{frame}\frametitle{Positive interest rate shock $v_{t}$ increases}

\begin{itemize}
\item Assume $1-\beta \delta _{\pi }<0$

\item $\Delta x=0$ shifts down and inflation and output gap fall

\item as the shock slowly disappears, $\Delta x=0$ shifts back up and output
and inflation return to long-run values

\item if shock were permanent, output and inflation would never return to
steady-state values

\item note output and inflation move together -- no policy tradeoffs between
keeping inflation on target and output on target if no shocks to
supply
\end{itemize}
\end{frame}


% \subsection{Cochrane's Critique}

% \subsubsection{Determinacy -- How can we rule out all unstable equilibria?}

% \begin{frame}
% \begin{itemize}
% \item Cochrane's answer is that the Fed promises to "blow up the economy" if
% price and output do not jump to allow non-explosive equilibrium

% \begin{itemize}
% \item Can the Fed blow up the economy? that is, make price and/or output
% explode?

% \begin{itemize}
% \item Output cannot reach $\pm \infty $

% \item Price can reach 0 or $\infty $

% \item Interest rate cannot go negative
% \end{itemize}
% \end{itemize}
% \end{itemize}
% \end{frame}

% \begin{frame}
% \begin{itemize}
% \item Cochrane views output trajectory as model flaw -- should price take
% off on an unstable trajectory with explosive output, the Calvo fairy would
% visit more often, as he does in Argentina

% \item Yes, the Fed can blow up the economy

% \item Would the Fed blow up the economy?

% \begin{itemize}
% \item Dynamic inconsistency -- optimal to promise to blow up the economy to
% prevent it from embarking on an unstable path

% \item Not likely to actually carry out promise

% \item More likely policy is that economy can get on unstable path and the
% Fed has plans to move it to a more favorable equilibrium

% \item Latter does not rule out the path
% \end{itemize}
% \end{itemize}
% \end{frame}

\subsubsection{New Keynesian Model does not have Old Keynesian dynamics}

\begin{frame}
\begin{itemize}
\item Monetary authority wants to offset positive demand shock

\item Old Keynesian dynamics

\begin{itemize}
\item $M\downarrow \rightarrow i \uparrow \rightarrow$ demand $%
\downarrow \rightarrow$with sticky prices $Y\downarrow $

\item over time $P\downarrow $ due to low demand, implying $\frac{M}{P}$
back up

\item model is stable, and economy returns to equilibrium over time
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}
\begin{itemize}
\item New Keynesian dynamics

\begin{itemize}
\item $i\uparrow \rightarrow Y$ and $\pi $ both jump down

\item once shock goes away, everything returns to equilibrium

\item no dynamics other than dynamics of shock

\item model is unstable, so economy always in equilibrium

\item if output and inflation do not jump correct amount, off on explosive
path
\end{itemize}
\end{itemize}
\end{frame}


\subsubsection{Identification}

\begin{frame}\frametitle{Taylor Rule}
\[
i_{t}=\delta _{\pi }\pi _{t}+\delta _{x}x_{t}+v_{t}
\]

\begin{itemize}
\item When $v_{t}$ increases, both $\pi _{t}$ and $x_{t}$ must jump to rule
out explosions

\begin{itemize}
\item Therefore error and rhs variables are correlated

\item There are no good instruments as need $i_{t}$ to respond to the
variables caused by the error
\end{itemize}

\item $\delta _{\pi }$ and $\delta _{x}$ do not show up in the dynamics of
the model
\item They appear in the roots

\item They need to be large enough for both roots to be unstable (requiring
the jump)
\end{itemize}
\end{frame}


\section{Monetary Transmission Mechanism}

\begin{frame}
\begin{itemize}
\item Write IS curve as a function of real interest rate gap%
\[
x_{t}=E_{t}x_{t+1}-\frac{1}{\sigma }\left( \hat{\imath}_{t}-E_{t}\pi
_{t+1}\right) +u_{t}
\]

\[
x_{t}=E_{t}x_{t+1}-\frac{1}{\sigma }\left( \hat{r}_{t}-\tilde{r}_{t}\right)
\hspace{0.3in}\text{ where }\tilde{r}_{t}=\sigma u_{t}
\]

\item $\tilde{r}_{t}$ is the Wicksellian real interest rate

\item solve forward%
\[
x_{t}=-\frac{1}{\sigma }\sum_{i=0}^{\infty }E_{t}\left( \hat{r}_{t+i}-\tilde{%
r}_{t+i}\right)
\]
\end{itemize}
\end{frame}

\begin{frame}
\begin{itemize}
\item output depends negatively on expected future real interest rate
deviations

\item if interest rates are always expected to equal the Wicksellian real
rate, then output gaps are zero

\item Where is money in the model?%
\[
\hat{m}_{t}-\hat{p}_{t}=\left( \frac{1}{bi^{ss}}\right) \left( \sigma \hat{y}%
_{t}-\hat{\imath}_{t}\right)
\]

\item money adjusts to get the desired interest rate
\end{itemize}
\end{frame}

\section{Economic Disturbances}

\begin{frame}\frametitle{Demand}

\begin{itemize}
\item Taste

\begin{itemize}
\item Amend utility function to have a taste shock ($\psi$)
\[
E_{t}\sum_{i=0}^{\infty }\beta ^{i}\left[ \left( \frac{\psi _{t+i}^{1-\sigma
}C_{t+i}^{1-\sigma }}{1-\sigma }\right) +\frac{\gamma }{1-b}\left( \frac{%
M_{t+i}}{P_{t+_{i}}}\right) ^{1-b}-\chi \frac{N_{t+i}^{1+\eta }}{1+\eta }%
\right]
\]

\item Euler equation becomes%
\[
\psi _{t}^{1-\sigma }C_{t}^{-\sigma }=\beta E_{t}\psi _{t+1}^{1-\sigma
}C_{t+1}^{-\sigma }\frac{\left( 1+i_{t}\right) }{1+\pi _{t+1}}
\]
\item Linearized around the zero inflation steady state%
\[
\hat{c}_{t}=E_{t}\hat{c}_{t+1}-\frac{1}{\sigma }\left( \hat{\imath}%
_{t}-E_{t}\pi _{t+1}\right) +\left( \frac{\sigma -1}{\sigma }\right) \left(
E_{t}\psi _{t+1}-\psi _{t}\right)
\]%
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Government spending}

\begin{itemize}
\item Resource constraint changes%
\[
\hat{y}_{t}=\left( \frac{C}{Y}\right) ^{ss}\hat{c}_{t}+\left( \frac{G}{Y}%
\right) ^{ss}\hat{g}_{t}
\]

\item IS curve becomes
\[
\hat{x}_{t}=E_{t}\hat{x}_{t+1}-\frac{1}{\tilde{\sigma}}\left( \hat{\imath}%
_{t}-E_{t}\pi _{t+1}\right) +\xi _{t}
\]%
\begin{eqnarray*}
\xi _{t} &=&\left( \frac{\sigma -1}{\sigma }\right) \left( \frac{C}{Y}%
\right) ^{ss}\left( E_{t}\psi _{t+1}-\psi _{t}\right) -\left( \frac{G}{Y}%
\right) ^{ss}E_{t}\left( \hat{g}_{t+1}-\hat{g}_{t}\right) \\
&&+\left( E_{t}\hat{y}_{t+1}^{f}-\hat{y}_{t}^{f}\right)
\end{eqnarray*}%
\[
\frac{1}{\tilde{\sigma}}=\frac{1}{\sigma }\left( \frac{C}{Y}\right) ^{ss}
\]

\item expected changes matter
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Supply}

\begin{itemize}
\item If no shocks to supply, then stabilization of inflation stabilizes
output gap

\item Solving New Keyneisan Phillips Curve forward yields%
\[
\pi _{t}=\kappa \sum_{t=0}^{\infty }\beta ^{i}E_{t}x_{t+i}
\]

\item Implies that if expected future output gaps are zero, then inflation
is at its target of zero

\item Supply shocks change this, raising inflation for each level of output
and producing policy tradeoffs

\item What might supply shocks be?
\end{itemize}
\end{frame}

\begin{frame}
\begin{itemize}
\item Equation determining inflation before linearization, contained tastes,
marginal costs, and productivity

\item Shocks to these variables can add shocks to the Phillips curve

\begin{itemize}
\item However, these shocks constitute shocks to the output gap evaluated at
changing values of full-employment output

\item Remember the output gap is evaluated relative to a fixed steady state,
so the equilibrium output gap responds to supply shocks

\item Monetary authority should be stabilizing output around the fluctuating
output gap, yielding no policy tradeoffs
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Sticky wages and prices}

\begin{itemize}
\item If only prices are sticky, optimal policy adjusts to keep prices from
ever having to adjust (zero inflation)

\item If wages are sticky, and there are real shocks, then a policy which
keeps prices from ever having to adjust could increase the output gap

\begin{itemize}
\item Negative productivity shock reduces the marginal product of labor

\item Rigid nominal wages and a monetary policy to keep prices from
adjusting would imply MPL$\left( N_{0}\right) <\frac{W}{P}.$ Firms would
reduce employment and output.

\item Now, there is a tradeoff between stabilizing inflation and the output
gap.
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Caplin and Spulber (1987)}

\begin{itemize}
\item The firms most likely to adjust prices are the ones with the most
suboptimal prices

\item The distribution of prices (relative to optimal) is stationary

\item If the initial distribution of prices is uniform, so is the stationary
distribution

\item A monetary shock changes the rate of adjustment, but not the
distribution or prices implying that aggregate prices are not sticky
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Conclusion}
\begin{itemize}
\item New Keynesian model:
\begin{enumerate}
\item Nominal rigidities: prices are sticky, i.e. slow to adjust.
\item This leads to a role for stabilization policies.
\end{enumerate}
\item Next time: financial frictions.
\end{itemize}
\end{frame}

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

\appendix

\begin{frame}
\begin{itemize}
\item trace is sum of roots
\begin{eqnarray*}
\theta _{1}+\theta _{2} &=&\frac{1}{\beta }+1+\frac{\beta \delta _{x}+\kappa
}{\beta \sigma } \\
&=&1+\frac{\delta _{x}}{\sigma }+\frac{1}{\beta }\left( 1+\frac{\kappa }{%
\sigma }\right)
\end{eqnarray*}

\item characteristic equation%
\[
\left[ \frac{1}{\beta }-\theta \right] \left[ 1+\frac{\beta \delta
_{x}+\kappa }{\beta \sigma }-\theta \right] +\frac{\kappa }{\beta }\left[
\frac{\beta \delta _{\pi }-1}{\sigma \beta }\right] =0
\]%
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Stability requirements}

\begin{itemize}
\item Since determinant and trace are positive, eigenvalues must be positive

\item Therefore, stability requirement is
\[
\left( \theta _{1}-1\right) \left( \theta _{2}-1\right) >0
\]

\item Multiplying out yields%
\begin{equation}
\theta _{1}\theta _{2}-\left( \theta _{1}+\theta _{2}\right) +1>0  \tag{SR}
\end{equation}

\item Determinant is%
\[
\theta _{1}\theta _{2}=\frac{1}{\beta }\left( 1+\frac{\delta _{x}}{\sigma }+%
\frac{\kappa \delta _{\pi }}{\sigma }\right)
\]
\end{itemize}
\end{frame}

\begin{frame}
\begin{itemize}
\item Trace is
\[
\theta _{1}+\theta _{2}=1+\frac{\delta _{x}}{\sigma }+\frac{1}{\beta }\left(
1+\frac{\kappa }{\sigma }\right)
\]

\item Substituting into stability requirements (SR)%
\[
\frac{1}{\beta }\left( 1+\frac{\delta _{x}}{\sigma }+\frac{\kappa \delta
_{\pi }}{\sigma }\right) -\left[ 1+\frac{\delta _{x}}{\sigma }+\frac{1}{%
\beta }\left( 1+\frac{\kappa }{\sigma }\right) \right] +1>0
\]

\item Simplifying%
\[
\delta _{x}\left( \frac{1}{\beta }-1\right) +\frac{\kappa }{\beta }\left(
\delta _{\pi }-1\right) >0
\]%
\[
\delta _{x}\left( 1-\beta \right) +\kappa \left( \delta _{\pi }-1\right) >0
\]%
Implying that the responses to the output gap and inflation must be large
enough
\end{itemize}
\end{frame}

\end{document}
