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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 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: 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: Go through basic NK model.
\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}

\begin{frame}\frametitle{Environment}
\begin{itemize}
\item Three agents:
  \begin{itemize}
  \item Household: Consume, work, save, hold money.
  \item Final goods producer: takes intermediate goods and produces consumable.
  \item Intermediate goods producer: Uses capital and tech, sells to final goods producer.
  \end{itemize}
\item Preferences and technology:
  \begin{itemize}
  \item Money in the utility function;
  \item Cobb-Douglas intermeidate production;
  \item CES aggregator for final good;
  \item Price rigidities generate market power for intermediate goods producer.
  \end{itemize}
\end{itemize}
\end{frame}

\section{Canonical NK Model}

\begin{frame}\frametitle{Household's Problem}
\begin{itemize}
\item Maximize expected utility which depends on a composite consumption
good, real money, and leisure%
\[
E_{t}\sum_{i=0}^{\infty }\beta ^{i}\left[ \left( \frac{C_{t+i}^{1-\sigma }}{%
1-\sigma }\right) +\frac{\gamma }{1-b}\left( \frac{M_{t+i}}{P_{t+_{i}}}%
\right) ^{1-b}-\chi \frac{N_{t+i}^{1+\eta }}{1+\eta }\right]
\]
s.t.
\[
C_{t}+\frac{M_{t}}{P_{t}}+\frac{B_{t}}{P_{t}}=\frac{W_{t}}{P_{t}}N_{t}+\frac{%
M_{t-1}}{P_{t}}+\frac{\left( 1+i_{t-1}\right) B_{t-1}}{P_{t}}+\Pi _{t}
\]%
\begin{itemize}
\item Upper case: nominal; lower case: real.
\item $\frac{M}{P}$: real money balances;
\item $\frac{B}{P}$: real bonds;
\item $\Pi$: firm profits (rigidities yield profits);
\item $i_{t}$: nominal interest rate.
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Household optimization problem}
\begin{itemize}
\item define%
\[
\omega _{t}=\frac{\left( m_{t-1}+\left( 1+i_{t-1}\right) b_{t-1}\right) }{%
1+\pi _{t}}=C_{t}+m_{t}+b_{t}-w_{t}N_{t}-\Pi _{t}
\]
\item $\omega$: real cash in hand.
\item $\pi$: rate of inflation.
\item substitute $b_{t}$ out by using%
\[
b_{t}=\omega _{t}-C_{t}-m_{t}+w_{t}N_{t}+\Pi _{t}
\]

\item value function%
\begin{align}
V\left( \omega _{t}\right) &=\max \left[ \left( \frac{C_{t}^{1-\sigma }}{%
1-\sigma }\right) +\frac{\gamma }{1-b}\left( \frac{M_{t}}{P_{t}}\right)
^{1-b}-\chi \frac{N_{t}^{1+\eta }}{1+\eta }\right] \nonumber\\&+\beta E_{t}V\left(
\omega _{t+1}\right)
\end{align}
\[
\omega _{t+1}=\frac{\left( m_{t}+\left( 1+i_{t}\right) \left( \omega
_{t}-C_{t}-m_{t}+w_{t}N_{t}+\Pi _{t}\right) \right) }{1+\pi _{t+1}}
\]
\end{itemize}
\end{frame}

\begin{frame}
\begin{itemize}
\item first order conditions

\[
C_{t}^{-\sigma }-\beta E_{t}V_{\omega }\left( \omega _{t+1}\right) \frac{%
\left( 1+i_{t}\right) }{1+\pi _{t+1}}=0
\]%
\[
\gamma m_{t}^{-b}-\beta E_{t}V_{\omega }\left( \omega _{t+1}\right) \frac{%
i_{t}}{1+\pi _{t+1}}=0
\]%
\[
-\chi N_{t}^{\eta }+\beta E_{t}V_{\omega }\left( \omega _{t+1}\right) w_{t}%
\frac{1+i_{t}}{1+\pi _{t+1}}=0
\]

\item using envelope condition
\[
V_{\omega }\left( \omega _{t}\right) =\beta E_{t}V_{\omega }\left( \omega
_{t+1}\right) \frac{1+i_{t}}{1+\pi _{t+1}}=C_{t}^{-\sigma }
\]
\end{itemize}
\end{frame}

\begin{frame}
\begin{itemize}
\item FO conditions become%
\begin{itemize}
\item Euler:
\[
C_{t}^{-\sigma }=\beta E_{t}C_{t+1}^{-\sigma }\frac{\left( 1+i_{t}\right) }{%
1+\pi _{t+1}}
\]%
\item Money holdings:
\[
\gamma m_{t}^{-b}=\beta E_{t}C_{t+1}^{-\sigma }\frac{i_{t}}{1+\pi _{t+1}}%
=C_{t}^{-\sigma }\frac{i_{t}}{1+i_{t}}
\]%
\item Labor:
\[
\chi N_{t}^{\eta }=\beta E_{t}C_{t+1}^{-\sigma }\frac{w_{t}\left(
1+i_{t}\right) }{1+\pi _{t+1}}=C_{t}^{-\sigma }w_{t}
\]%
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Final Goods Producer}
\begin{itemize}
\item Composite consumption good is CES%
\[
C_{t}=\left[ \int_{0}^{1}c_{jt}^{\frac{\theta -1}{\theta }}dj\right] ^{\frac{%
\theta }{\theta -1}}\hspace{0.5in}\theta >1
\]
\item Choose $c_{jt}$ to minimize the cost of buying $C_{t}$%
\[
\min_{c_{jt}}\int_{0}^{1}p_{jt}c_{jt}dj
\]%
subject to%
\[
\left[ \int_{0}^{1}c_{jt}^{\frac{\theta -1}{\theta }}dj\right] ^{\frac{%
\theta }{\theta -1}}\geq C_{t}
\]

\end{itemize}
\end{frame}


\begin{frame}
  \begin{itemize}
\item set up a Lagrangian%
\[
L=\int_{0}^{1}p_{jt}c_{jt}dj+\Psi _{t}\left[ C_{t}-\left(
\int_{0}^{1}c_{jt}^{\frac{\theta -1}{\theta }}dj\right) ^{\frac{\theta }{%
\theta -1}}\right]
\]
\item First order condition with respect to $c_{jt}$
\[
p_{jt}-\Psi _{t}\frac{\theta }{\theta -1}\left( \int_{0}^{1}c_{jt}^{\frac{%
\theta -1}{\theta }}dj\right) ^{\frac{\theta }{\theta -1}-1}\frac{\theta -1}{%
\theta }c_{jt}^{\frac{-1}{\theta }}=0
\]%
\[
p_{jt}-\Psi _{t}\left( \int_{0}^{1}c_{jt}^{\frac{\theta -1}{\theta }%
}dj\right) ^{\frac{1}{\theta -1}}c_{jt}^{\frac{-1}{\theta }}=0
\]%
\[
p_{jt}-\Psi _{t}C_{t}^{\frac{1}{\theta }}c_{jt}^{\frac{-1}{\theta }}=0
\]%
\[
c_{jt}=\left( \frac{p_{jt}}{\Psi _{t}}\right) ^{-\theta }C_{t}
\]
\end{itemize}
\end{frame}
\begin{frame}
  \begin{itemize}
\item Integrate over $j$ to solve for $C_{t}$ and eliminate the multiplier%
\[
C_{t}=\left[ \int_{0}^{1}\left( \left( \frac{p_{jt}}{\Psi _{t}}\right)
^{-\theta }C_{t}\right) ^{\frac{\theta -1}{\theta }}dj\right] ^{\frac{\theta
}{\theta -1}}=\Psi _{t}^{\theta }C_{t}\left[ \int_{0}^{1}p_{jt}^{1-\theta }dj%
\right] ^{\frac{\theta }{\theta -1}}
\]

\item Solve for $\Psi _{t}$%
\[
1=\Psi _{t}\left[ \int_{0}^{1}p_{jt}^{1-\theta }dj\right] ^{\frac{1}{\theta
-1}}
\]%
\[
\Psi _{t}=\left[ \int_{0}^{1}p_{jt}^{1-\theta }dj\right] ^{\frac{1}{1-\theta
}}\equiv P_{t}
\]%
\item We have an expression for the aggregate price.
\end{itemize}
\end{frame}

\begin{frame}
Substituting into demand yields demand as a function of relative
price and of composite consumption%
\[
c_{jt}=\left( \frac{p_{jt}}{P_{t}}\right) ^{-\theta }C_{t}
\]%
the price elasticity of demand is $\theta >1$

\begin{itemize}
\item Important: downward-sloping demand for each good j $\rightarrow$ intermediate good producers have pricing power.
\end{itemize}
\end{frame}



\begin{frame}\frametitle{Firm's problem}
\begin{itemize}
\item Firms maximize profits subject to constraints

\begin{itemize}
\item Constraints

\begin{itemize}
\item production function%
\[
c_{jt}=Z_{t}N_{jt}\hspace{0.5in}EZ_{t}=1
\]

\item demand curve%
\[
c_{jt}=\left( \frac{p_{jt}}{P_{t}}\right) ^{-\theta }C_{t}
\]

\item with probability $1 - \omega$ Calvo fairy arrives and firm can adjust price.
\end{itemize}

\item choose labor to minimize costs taking the real wage as given
\[
\min w_{t}N_{jt}+\varphi _{t}\left( c_{jt}-Z_{t}N_{jt}\right)
\]
\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}
\begin{itemize}
\item first order condition
\[
\varphi _{t}=\frac{w_{t}}{Z_{t}}=\frac{w_{t}}{MPN}=\text{real marginal cost. common to all.}
\]

\item Next: choose price $\left( p_{jt}\right) $ to maximize real discounted
profits%
\[
E_{t}\sum_{i=0}^{\infty }\omega ^{i}\Delta _{i,t+i}\left[ \frac{p_{jt}}{%
P_{t+i}}c_{jt+i}-\varphi _{t+i}c_{jt+i}\right]
\]%
where discount factor is
\[
\Delta _{i,t+i}=\beta ^{i}\left( \frac{C_{t+i}}{C_{t}}\right) ^{-\sigma }
\]%
\end{itemize}
\end{frame}


\begin{frame}
\begin{itemize}
\item since $c_{jt}$ depends on price through demand, substitute demand
curve for $c_{jt}=\left( \frac{p_{jt}}{P_{t}}\right) ^{-\theta }C_{t}$%
\[
\max_{p_{jt}}E_{t}\sum_{i=0}^{\infty }\omega ^{i}\Delta _{i,t+i}\left[ \left( \frac{p_{jt}%
}{P_{t+i}}\right) ^{1-\theta }-\varphi _{t+i}\left( \frac{p_{jt}}{P_{t+i}}%
\right) ^{-\theta }\right] C_{t+i}
\]

\item optimal $p_{jt}=p_{t}^{\ast }$ since {\color{red}firms are identical in all ways
except date at which last changed price}: FOC with respect to $p_{jt}$
\begin{align*}\hspace{-5mm}
E_{t}\sum_{i=0}^{\infty }\omega ^{i}\Delta _{i,t+i}\left[ \left( 1-\theta
\right) \left( \frac{p_{jt}}{P_{t+i}}\right) ^{-\theta }+\varphi
_{t+i}\theta \left( \frac{p_{jt}}{P_{t+i}}\right) ^{-\theta -1}\right] \frac{%
C_{t+i}}{P_{t+i}}=0
\end{align*}
\begin{align*}\hspace{-5mm}
E_{t}\sum_{i=0}^{\infty }\omega ^{i}\Delta _{i,t+i}\left[ \left( 1-\theta
\right) \left( \frac{p_{t}^{\ast }}{P_{t+i}}\right) +\varphi _{t+i}\theta %
\right] \left( \frac{1}{p_{t}^{\ast }}\right) \left( \frac{p_{t}^{\ast }}{%
P_{t+i}}\right) ^{-\theta }C_{t+i}=0
\end{align*}
\end{itemize}
\end{frame}

\begin{frame}
\begin{itemize}
\item first term%
\begin{eqnarray*}
&&E_{t}\sum_{i=0}^{\infty }\omega ^{i}\Delta _{i,t+i}\left( 1-\theta \right)
\frac{p_{t}^{\ast }}{P_{t+i}}P_{t+i}^{\theta }C_{t+i} \\
&=&\frac{p_{t}^{\ast }}{P_{t}}\left( 1-\theta \right)
E_{t}\sum_{i=0}^{\infty }\omega ^{i}\Delta _{i,t+i}P_{t}P_{t+i}^{\theta
-1}C_{t+i}
\end{eqnarray*}

\item second term%
\[
E_{t}\sum_{i=0}^{\infty }\omega ^{i}\Delta _{i,t+i}\varphi _{t+i}\theta
P_{t+i}^{\theta }C_{t+i}=\theta E_{t}\sum_{i=0}^{\infty }\omega ^{i}\Delta
_{i,t+i}\varphi _{t+i}P_{t+i}^{\theta }C_{t+i}
\]
\end{itemize}
\end{frame}


\begin{frame}
\begin{itemize}
\item solve for relative price%
\begin{eqnarray*}
\frac{p_{t}^{\ast }}{P_{t}} &=&\left( \frac{\theta }{\theta -1}\right) \frac{%
E_{t}\sum_{i=0}^{\infty }\omega ^{i}\Delta _{i,t+i}\varphi
_{t+i}P_{t+i}^{\theta }C_{t+i}}{E_{t}\sum_{i=0}^{\infty }\omega ^{i}\Delta
_{i,t+i}P_{t}P_{t+i}^{\theta -1}C_{t+i}} \\
&=&\left( \frac{\theta }{\theta -1}\right) \frac{E_{t}\sum_{i=0}^{\infty
}\omega ^{i}\Delta _{i,t+i}\varphi _{t+i}\left( \frac{P_{t+i}}{P_{t}}\right)
^{\theta }C_{t+i}}{E_{t}\sum_{i=0}^{\infty }\omega ^{i}\Delta _{i,t+i}\left(
\frac{P_{t+i}}{P_{t}}\right) ^{\theta -1}C_{t+i}}
\end{eqnarray*}%
where last equality multiplies numerator and denominator by $P_{t}^{-\theta
} $
\item $\varphi$: real marginal cost (of labor).
\item use%
\[
\Delta _{i,t+i}C_{t+i}=\beta ^{i}C_{t+i}^{1-\sigma }C_{t}^{\sigma }
\]%
\[
\frac{p_{t}^{\ast }}{P_{t}}=\left( \frac{\theta }{\theta -1}\right) \frac{%
E_{t}\sum_{i=0}^{\infty }\omega ^{i}\beta ^{i}\varphi _{t+i}\left( \frac{%
P_{t+i}}{P_{t}}\right) ^{\theta }C_{t+i}^{1-\sigma }}{E_{t}\sum_{i=0}^{%
\infty }\omega ^{i}\beta ^{i}\left( \frac{P_{t+i}}{P_{t}}\right) ^{\theta
-1}C_{t+i}^{1-\sigma }}
\]
\item Price setting equation for intermediate goods producers.
\end{itemize}
\end{frame}

\begin{frame}\frametitle{Flexible price equilibrium}
\begin{itemize}
\item every firm adjusts every period, so $%
\omega =0;$ lose all but first term%
\[
\frac{p_{t}^{\ast }}{P_{t}}=\left( \frac{\theta }{\theta -1}\right) \varphi
_{t}=\mu \varphi _{t}
\]

\begin{itemize}
\item implying that price is a markup over marginal costs since $\theta >1$

\item {\color{red}since price exceeds marginal cost, output is inefficiently low}

\item since all firms charge the same price%
\[
\varphi =\frac{1}{\mu }
\]

\item firms choose labor such that%
\[
\frac{Z_{t}}{\mu }=w_{t}
\]
\end{itemize}
\end{itemize}
\end{frame}



\begin{frame}
\begin{itemize}
\item households choose labor such that
\[
\frac{\chi N_{t}^{\eta }}{C_{t}^{-\sigma }}=w
\]

\item in flexible price equilibrium
\[
C_{t}=Y_{t}=Z_{t}N_{t}=\left( \frac{1}{\chi \mu }\right) ^{\frac{1}{\sigma
+\eta }}Z_{t}^{\frac{1+\eta }{\sigma +\eta }}
\]

\begin{itemize}
\item Full employment output is potentially affected by shocks to

\begin{itemize}
\item productivity $\left( Z_{t}\right) $

\item tastes $\left( \chi \right) $

\item demand elasticity (markup) $\left( \mu \right) $\pagebreak
\end{itemize}
\end{itemize}
\end{itemize}
\end{frame}


\begin{frame}\frametitle{Thinking about sticky prices}
\begin{itemize}
\item Return to sticky prices

\begin{itemize}
\item recall
\[
\left[ \int_{0}^{1}p_{jt}^{1-\theta }dj\right] ^{\frac{1}{1-\theta }}\equiv
P_{t}
\]%
\[
\left[ \int_{0}^{1}p_{jt}^{1-\theta }dj\right] =P_{t}^{1-\theta }
\]

\item Aggregate price level%
\[
P_{t}^{1-\theta }=\left( 1-\omega \right) \left( p_{t}^{\ast }\right)
^{1-\theta }+\omega P_{t-1}^{1-\theta }
\]

\begin{itemize}
\item $1-\omega $ firms adjust this period and charge the optimal price

\item $\omega $ do not adjust, and since the adjusting firms are drawn
randomly, the price level for non-adjusters is unchanged\pagebreak
\end{itemize}
\end{itemize}
\end{itemize}
\end{frame}


\section{Linearized New Keynesian Model}

\begin{frame}\frametitle{New Keynesian Phillips Curve}
  \begin{itemize}
  \item Phillips Curve: rel. btwn. inflation (expected) and unemployment (output gap).
  \item Intermediate price setting:
\[
\frac{p_{t}^{\ast }}{P_{t}}=\left( \frac{\theta }{\theta -1}\right) \frac{%
E_{t}\sum_{i=0}^{\infty }\omega ^{i}\beta ^{i}\varphi _{t+i}\left( \frac{%
P_{t+i}}{P_{t}}\right) ^{\theta }C_{t+i}^{1-\sigma }}{E_{t}\sum_{i=0}^{%
\infty }\omega ^{i}\beta ^{i}\left( \frac{P_{t+i}}{P_{t}}\right) ^{\theta
-1}C_{t+i}^{1-\sigma }}
\]%
\item Agg. price setting:
\[
P_{t}^{1-\theta }=\left( 1-\omega \right) \left( p_{t}^{\ast }\right)
^{1-\theta }+\omega P_{t-1}^{1-\theta }
\]
  \end{itemize}
\begin{itemize}
\item Let the relative price the firm chooses when he adjusts be
\[
Q_{t}=\frac{p_{t}^{\ast }}{P_{t}}
\]

\item $Q=1$ in steady state and when all firms can adjust every period
\end{itemize}
\end{frame}



\begin{frame}\frametitle{Tedious algebra later...}
\begin{itemize}
\item Vars w/o time subscript are steady-state ($C$)
\item Hats: deveiation from steady-state.
\item We can derive the price that would be set by the firm:
\[
\rightarrow\hat{p}_{t}^{\ast }=\hat{q}_{t}+\hat{p}_{t}=\left( 1-\omega \beta \right)
\sum_{i=0}^{\infty }\omega ^{i}\beta ^{i}\left[ E_{t}\left( \hat{\varphi}%
_{t+i}+\hat{p}_{t+i}\right) \right]
\]%
the optimal nominal price equals the expected discounted value of future
nominal ($\omega$) marginal costs
\item marginal costs: real marginal cost of labor + price.
\item can write equation as%
\[
\omega \beta E_{t}\left( \hat{q}_{t+1}+\hat{p}_{t+1}\right) =\hat{q}_{t}+%
\hat{p}_{t}-\left( 1-\omega \beta \right) \left( \hat{\varphi}_{t}+\hat{p}%
_{t}\right)
\]%
\item $\hat{q}$: deviation from SS $p^{*}/P^{*}$. $\hat{p}$: deviation from SS $p^{*}$
\end{itemize}
\end{frame}



\begin{frame}
\begin{itemize}
Solving for $\hat{q}_{t}$%
\[
\hat{q}_{t}=\left( 1-\omega \beta \right) \hat{\varphi}_{t}+\omega \beta %
\left[ E_{t}\left( \hat{q}_{t+1}+\hat{p}_{t+1}\right) -\hat{p}_{t}\right]
\]%
\[
\hat{q}_{t}=\left( 1-\omega \beta \right) \hat{\varphi}_{t}+\omega \beta
E_{t}\left( \hat{q}_{t+1}+\pi _{t+1}\right)
\]

\item using $\hat{q}_{t}=\frac{\omega }{1-\omega }\hat{\pi}_{t}$ to
eliminate $\hat{q}_{t}$%
\[
\frac{\omega }{1-\omega }\pi _{t}=\left( 1-\omega \beta \right) \hat{\varphi}%
_{t}+\omega \beta E_{t}\left( \frac{\omega }{1-\omega }\pi _{t+1}+\pi
_{t+1}\right)
\]%
\[
\pi _{t}=\frac{\left( 1-\omega \beta \right) \left( 1-\omega \right) \hat{%
\varphi}_{t}}{\omega }+\beta E_{t}\pi _{t+1}
\]%
\[
\pi _{t}=\tilde{\kappa}\hat{\varphi}_{t}+\beta E_{t}\pi _{t+1}
\]

\item Recall: $\varphi$: roughly measure of the output gap (proportional to price ratio).
\end{itemize}
\end{frame}



\begin{frame}
\begin{itemize}
\item no backward-looking terms, expected future inflation matters, not
lagged inflation (recall Lucas Critique)

\item marginal cost instead of output gap -- under some restrictions the same

\begin{itemize}
\item from household's labor supply decision, real wage must equal marginal
rate of substitution between leisure and consumption%
\[
\hat{w}_{t}-\hat{p}_{t}=\eta \hat{n}_{t}+\sigma \hat{y}_{t}
\]

\item using%
\[
\hat{c}_{t}=\hat{y}_{t}=\hat{n}_{t}+\hat{z}_{t}
\]%
\end{itemize}
\end{itemize}
\end{frame}


\begin{frame}
\begin{itemize}
\item marginal costs equals real wage divided by marginal product of labor $%
\left( Z_{t}\right) $%
\begin{eqnarray*}
\hat{\varphi}_{t} &=&\hat{w}_{t}-\hat{p}_{t}-\hat{z}_{t}=\hat{w}_{t}-\hat{p}%
_{t}-\left( \hat{y}_{t}-\hat{n}_{t}\right) \\
&=&\eta \hat{n}_{t}+\sigma \hat{y}_{t}-\hat{z}_{t}=\eta \left( \hat{y}_{t}-%
\hat{z}_{t}\right) +\sigma \hat{y}_{t}-\hat{z}_{t} \\
&=&\left( \eta +\sigma \right) \left[ \hat{y}_{t}-\frac{1+\eta }{\left( \eta
+\sigma \right) }\hat{z}_{t}\right]
\end{eqnarray*}%
where%
\[
\hat{y}_{t}^{f}=\frac{1+\eta }{\left( \eta +\sigma \right) }\hat{z}_{t}
\]%
implying that
\[
\hat{\varphi}_{t}=\left( \eta +\sigma \right) \left[ \hat{y}_{t}-\hat{y}%
_{t}^{f}\right] =\gamma \left[ \hat{y}_{t}-\hat{y}_{t}^{f}\right]
\]

\item New Keynesian Phillips Curve becomes%
\[
\pi _{t}=\kappa x_{t}+\beta E_{t}\pi _{t+1}
\]
\item more complicated when do not have constant returns to scale, but
principle is same
\end{itemize}
\end{frame}



\begin{frame}\frametitle{IS curve}
\[
\hat{y}_{t}=E_{t}\hat{y}_{t+1}-\frac{1}{\sigma }\left( \hat{\imath}%
_{t}-E_{t}\pi _{t+1}\right)
\]

\begin{itemize}
\item expressed in terms of the 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%
\[
\hat{\imath}_{t}=\delta _{\pi }\pi _{t}+\delta _{x}x_{t}+v_{t}
\]
\item Start from this point next time.
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Conclusion}
\begin{itemize}
\item Today: New Keynsian Model
\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: More New Keynesian Models.
\end{itemize}
\end{frame}

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

\appendix

\begin{frame}\frametitle{Log-linearizing}
\begin{itemize}
\item dividing second equation by $P_{t}^{1-\theta }$%
\[
1=\left( 1-\omega \right) Q_{t}^{1-\theta }+\omega \left( \frac{P_{t-1}}{%
P_{t}}\right) ^{1-\theta }
\]

\item expressed in percent deviations about steady state with $\frac{P_{t-1}%
}{P_{t}}=1$
\[
1=\left( 1-\omega \right) \left( 1+\left( 1-\theta \right) \hat{q}%
_{t}\right) +\omega \left( 1-\left( 1-\theta \right) \hat{\pi}_{t}\right)
\]%
\[
\hat{q}_{t}=\frac{\omega }{1-\omega }\hat{\pi}_{t}
\]

\item rewrite first equation as (multiply through denominator)%
\begin{align*}\hspace{-5mm}
Q_{t}E_{t}\sum_{i=0}^{\infty }\omega ^{i}\beta ^{i}\left( \frac{P_{t+i}}{%
P_{t}}\right) ^{\theta -1}C_{t+i}^{1-\sigma }=\mu E_{t}\sum_{i=0}^{\infty
}\omega ^{i}\beta ^{i}\varphi _{t+i}\left( \frac{P_{t+i}}{P_{t}}\right)
^{\theta }C_{t+i}^{1-\sigma }
\end{align*}
\end{itemize}
\end{frame}



\begin{frame}
\begin{itemize}
\item $Q = \frac{p_{t}^{\ast }}{P_{t}}=\left( \frac{\theta }{\theta -1}\right) \frac{%
E_{t}\sum_{i=0}^{\infty }\omega ^{i}\beta ^{i}\varphi _{t+i}\left( \frac{%
P_{t+i}}{P_{t}}\right) ^{\theta }C_{t+i}^{1-\sigma }}{E_{t}\sum_{i=0}^{%
\infty }\omega ^{i}\beta ^{i}\left( \frac{P_{t+i}}{P_{t}}\right) ^{\theta
-1}C_{t+i}^{1-\sigma }}$
\item Approximate the left hand side as%
\begin{eqnarray*}
&&\frac{C^{1-\sigma }}{1-\omega \beta }+\left( \frac{C^{1-\sigma }}{1-\omega
\beta }\right) \hat{q}_{t} \\
&&+C^{1-\sigma }\sum_{i=0}^{\infty }\omega ^{i}\beta ^{i}\left[ \left(
1-\sigma \right) E_{t}\hat{C}_{t+i}+\left( \theta -1\right) \left( E_{t}\hat{%
p}_{t+i}-\hat{p}_{t}\right) \right]
\end{eqnarray*}
\item Vars w/o time subscript are steady-state ($C$)
\item Hats: deveiation from steady-state.
\item Approximate the right hand side as%
\begin{eqnarray*}
&&\mu \left( \frac{C^{1-\sigma }}{1-\omega \beta }\right) \varphi \\
&&+\mu \varphi C^{1-\sigma }\sum_{i=0}^{\infty }\omega ^{i}\beta ^{i}\left[
\left( 1-\sigma \right) E_{t}\hat{C}_{t+i}+\theta \left( E_{t}\hat{p}_{t+i}-%
\hat{p}_{t}\right) +E_{t}\hat{\varphi}_{t+i}\right]
\end{eqnarray*}
\end{itemize}
\end{frame}

\begin{frame}
\begin{itemize}
\item Equating and noting that $\mu \varphi =1$%
\[
\left( \frac{1}{1-\omega \beta }\right) \hat{q}_{t}+\sum_{i=0}^{\infty
}\omega ^{i}\beta ^{i}\left[ \left( -1\right) \left( E_{t}\hat{p}_{t+i}-\hat{%
p}_{t}\right) \right] =\sum_{i=0}^{\infty }\omega ^{i}\beta ^{i}\left[ E_{t}%
\hat{\varphi}_{t+i}\right]
\]%
\[
\rightarrow\left( \frac{1}{1-\omega \beta }\right) \hat{q}_{t}=\sum_{i=0}^{\infty
}\omega ^{i}\beta ^{i}\left[ E_{t}\left( \hat{\varphi}_{t+i}+\hat{p}%
_{t+i}\right) \right] -\left( \frac{1}{1-\omega \beta }\right) \hat{p}_{t}
\]%
\[
\rightarrow\hat{p}_{t}^{\ast }=\hat{q}_{t}+\hat{p}_{t}=\left( 1-\omega \beta \right)
\sum_{i=0}^{\infty }\omega ^{i}\beta ^{i}\left[ E_{t}\left( \hat{\varphi}%
_{t+i}+\hat{p}_{t+i}\right) \right]
\]%
the optimal nominal price equals the expected discounted value of future
nominal marginal costs

\item can write equation as%
\[
\omega \beta E_{t}\left( \hat{q}_{t+1}+\hat{p}_{t+1}\right) =\hat{q}_{t}+%
\hat{p}_{t}-\left( 1-\omega \beta \right) \left( \hat{\varphi}_{t}+\hat{p}%
_{t}\right)
\]%
\end{itemize}
\end{frame}

\end{document}
