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\title{\vspace{1.6cm} \\
	{\LARGE Lecture 7}\\[1ex]
{\LARGE New Keynesian Model}\\
	\noindent \hspace{-0.3cm} \rule{12.8cm}{0.6pt} \vspace{-0.7cm}}
\author{Macroeconomics EC2B1\\[3ex]  Benjamin Moll \\\vspace{0.5cm}}
\institute{London School of Economics, Winter 2024}





\begin{document}

%%%%%%%%%%%%%%%%%%

{
\setbeamertemplate{footline}{} 
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\titlepage
\end{frame}
}

\begin{frame}{New Keynesian Model}
\vspace{-3mm}
\bi
\item In a nutshell: \tr{New Keynesian model = RBC model with sticky prices}
\medskip
\item Simple framework to think about relationship between monetary policy, inflation and the business cycle
\medskip
\item RBC model: cannot even think about these issues!
\bi
\smallskip
\item real variables are completely separate from nominal variables (``monetary neutrality", ``classical dichotomy")
\smallskip
\item corollary: monetary policy has no effect on any real variables
\ei
\medskip
\item Sticky prices \tr{break ``monetary neutrality"}
\medskip
\item Sticky prices also \tr{break 1st welfare thm} $\Rightarrow$ rationale for stabilization policy
\medskip
\item New Keynesian model is \tr{current workhorse model} at central banks
\medskip
\item Some reason to believe that \tr{``demand shocks"} (e.g. consumer confidence, ``animal spirits") may drive business cycle
\bi
\smallskip
\item sticky prices = one way to get this story off the ground
\ei
\ei
\end{frame}


\begin{frame}{Why ``New" and ``Keynesian"?}
\tr{``New"} = methodological
\bi
\medskip
\item microfounded 
\medskip
\item in contrast to ``Old" Keynesian cross, IS-LM (or IS-MP-PC) and Keynesian large-scale macroeconometric models we discussed in Lecture 6
\ei 
\bigskip
\tr{``Keynesian"} = substantive
\bi
\medskip
\item like in ``Old" Keynesian theories, aggregate demand matters, stabilization policy can be desirable
\medskip
\item in contrast to RBC model
\ei 
\bigskip
Also see discussion in Kurlat, chapter 14.1
\end{frame}


\begin{frame}{New Keynesian models at the U.S. Fed and ECB}
\vspace{-3mm}
U.S. Fed EDO model and ECB New Area Wide Model (NAWM)
\begin{figure}[tph]
	\centering
\includegraphics[width=.5\textwidth]{figures/FRBUS_EDO}  \includegraphics[width=.35\textwidth]{figures/ECB_NAWM}
\end{figure}
\vspace{-1mm}
{\tiny \url{https://www.federalreserve.gov/econres/edo-models-about.htm} \& \url{https://www.ecb.europa.eu/pub/pdf/scpwps/ecbwp944.pdf}}\\[-1ex] {\tiny Also see \url{https://www.richmondfed.org/publications/research/econ_focus/2018/q2/federal_reserve}}
\end{frame}


\begin{frame}{What we'll cover}
\bi
\item Standard New Keynesian model used in academia and central banks is beyond scope of course
\bi
\medskip
\item dynamic stochastic general equilibrium (DSGE) model: like fully-fledged RBC model from last lecture but with sticky prices
\medskip
\item see e.g. Gali textbook on slide with readings
\ei
\bigskip
\item We will instead cover two-period version due to Mankiw and Weinzierl
\bigskip
\item Most insights and policy implications similar to those of standard New Keynesian model
\ei
\end{frame}



\begin{frame}{New Keynesian models used by central banks are much more complicated but basic structure \& logic are the same}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.65\textwidth]{figures/EDO_model_overview} 
\end{figure}
\vspace{-3mm}
{\tiny Chung, Kiley, Laforte ``Documentation of the Estimated, Dynamic, Optimization-based
(EDO) Model of the U.S. Economy: 2010 Version" \url{https://www.federalreserve.gov/econres/feds/documentation-of-the-estimated-dynamic-optimization-based-edo-model-of-the-us-economy-2010-version.htm}}
\end{frame}



\begin{frame}{{\large Effects of interest rate hike in U.S. Fed's own New Keynesian model -- our version will feature similar transmission mechanism}}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.6\textwidth]{figures/EDO_IRF} 
\end{figure}
\vspace{-3mm}
{\tiny Chung, Kiley, Laforte ``Documentation of the Estimated, Dynamic, Optimization-based
(EDO) Model of the U.S. Economy: 2010 Version" \url{https://www.federalreserve.gov/econres/feds/documentation-of-the-estimated-dynamic-optimization-based-edo-model-of-the-us-economy-2010-version.htm}}
\end{frame}


\begin{frame}{Plan}
\be
\item Preview: key diagram (aggregate demand \& supply)
\bigskip
\item Introducing money and inflation into the two-period RBC model
\bigskip
\item Flexible prices: monetary neutrality
\bigskip
\item Sticky prices: monetary non-neutrality
\ee
\bigskip
Next lecture: policy in the New Keynesian model
\end{frame}


\begin{frame}{Readings and supplementary materials}
\vspace{-4mm}
\be
\item Supplement on moodle: write-up of model including all the derivations
\smallskip
\item Mankiw \& Weinzierl (2011), ``An Exploration of Optimal Stabilization Policy"
\bi
\smallskip
\item will often abbreviate as ``MW"
\smallskip
\item read sections I-IV and IX-X (skip sections V-VIII)
\ei
\medskip
\item EC1P1 Lectures 7 and 8 (parts on money, inflation, quantity theory)
\medskip
\item Jones, Part 3 ``The Short Run", in particular chapters 11 to 13
\medskip
\item Section ``Monetary Policy: What Is the Best Evidence We Have?" in Nakamura and Steinsson (2018) ``Identification in Macroeconomics" \\[-1ex]
{\scriptsize \url{https://eml.berkeley.edu/~enakamura/papers/macroempirics.pdf}}
\medskip
\item Further readings for the interested (not examinable)
{\small
\bi
\smallskip
\item Wikipedia {\scriptsize \url{https://en.wikipedia.org/wiki/New_Keynesian_economics}}
\smallskip
\item Gali's book ``Monetary Policy, Inflation, and the Business Cycle"
\smallskip
\item Clarida, Gali and Gertler (1999), ``The Science of Monetary Policy: A New Keynesian Perspective"
\ei}
\ee
\end{frame}

\begin{frame}
\renewcommand{\baselinestretch}{1}
\vspace{3cm}
\begin{center}
{\huge Preview: Key Diagram}
\end{center}
\end{frame}

\begin{frame}{Key diagram: aggregate demand and aggregate supply}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.8\textwidth]{figures/lec7_1.eps} 
\end{figure}
\end{frame}



\begin{frame}{{\large Preview: with sticky prices, a negative demand shock causes recession}}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.8\textwidth]{figures/lec7_2.eps} 
\end{figure}
\end{frame}


\begin{frame}{Preview: monetary or fiscal policy can counteract recession}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.8\textwidth]{figures/lec7_5.eps} 
\end{figure}
\end{frame}


\begin{frame}
\renewcommand{\baselinestretch}{1}
\vspace{3cm}
\begin{center}
{\huge Introducing Money and Inflation}
\end{center}
\end{frame}


\begin{frame}{Starting point: competitive equilibrium in baby RBC model}
\vspace{-3mm}
\tr{\textbf{Definition:}} a competitive equilibrium are quantities $(C_1,C_2,I_1,K_2,Y_1,Y_2)$ and an interest rate $r_1$ such that
\be
\medskip
\item Utility maximization: taking as given $r_1$ and $W$, households choose $(C_1,C_2)$ to solve
$$\max_{C_1,C_2} \ U(C_1) + \beta U(C_2) \quad \mbox{s.t.} \quad C_1 + \frac{C_2}{1+r_1} = W$$
where $W$ is the PDV of firm profits (because households own firms)
\medskip
\item Profit maximization: firms maximize $W = \Pi_1 + \frac{\Pi_2}{1+r_1}$ or equivalently
$$W = \max_{K_2} \ \left\{A_1 K_1 - I_1 + \frac{A_2 K_2}{1+r_1} \right\}, \quad K_2 = I_1, \ Y_1 = A_1 K_1, \ Y_2 = A_2 K_2 $$
\item Market clearing: demand = supply for goods
\begin{align*}
\mbox{goods in period 1:} \quad &C_1 + I_1 = Y_1\\
\mbox{goods in period 2:} \quad&C_2 = Y_2
\end{align*}
\ee
\end{frame}

\begin{frame}{Reintroducing Nominal Prices}
\vspace{-3mm}
\bi
\item So far: simply set price of final consumption goods $P_1=P_2=1$
\bi
\medskip
\item without loss of generality: express price $r_1$ in terms of units consumption good (e.g. apples)
\ei
\medskip
\item Now: reintroduce nominal prices
\medskip
\item Example: instead of writing household budget constraint as
$$C_1 + \frac{C_2}{1+r_1} = \Pi_1 + \frac{\Pi_2}{1+r_1} $$
we now write
$$P_1 C_1 + \frac{P_2 C_2}{1+i_1} = P_1\Pi_1 + \frac{P_2\Pi_2}{1+i_1}$$
where $i_1$ = \tr{nominal interest rate}, i.e. in terms of \$ rather than apples
\medskip
\item For reasons that will become clear: also introduce ``period 0"
\bi
\smallskip
\item reference price $P_0$ determined some time before period 1, before households and firms know economic conditions like $A_1,A_2$ etc
\ei
\ei
\end{frame}



\begin{frame}{Inflation, real interest rate, Fisher equation}
\vspace{-3mm}
\bi
\item Definition: the \tr{inflation rate} is
$$\pi_2 = \frac{P_2-P_1}{P_1} \quad \mbox{and} \quad \pi_1 = \frac{P_1 - P_0}{P_0}  $$
\item Note: main role of ``pre-period" $0$ = being able to define $\pi_1$
\medskip
\item Definition: \tr{real interest rate} is nominal interest rate adjusted for inflation
$$1+r_1 = \frac{1 + i_1}{P_2/P_1} = \frac{1+i_1}{1+\pi_2}$$
\item Useful approximation when $r_1 \pi_2$ is small:
\begin{equation}\label{eq:Fisher}\tag{$\ast$}
\tr{i_1 \approx r_1 + \pi_2}
\end{equation}
\item Derivation of approximation
$$1+i_1 = (1+r_1)(1+\pi_2) = 1 + r_1 + \pi_2 + \underbrace{r_1 \pi_2}_{\approx 0} \approx 1 + r_1 + \pi_2  $$
\vspace{-5mm}
\item \eqref{eq:Fisher} is called \tr{``Fisher equation"}
\ei
\end{frame}


%\begin{frame}{Introducing money: money supply and demand}
%\vspace{-2mm}
%\bi
%\item Price level $P_t$  will end up being determined by monetary policy and equilibrium in the money market
%\bi
%\medskip
%\item money demand = money supply, $M_t^D = M_t^S$
%\medskip
%\item will sometimes drop $D$ and $S$ superscripts and just write $M_t$
%\medskip
%\item note: $M_t$ = \tr{stock} of money
%\ei
%\bigskip
%\item Money supply: central bank (monetary policy) sets money supply $M_t^S$
%\bigskip
%\item Money demand: follow Mankiw-Weinzierl and assume (see p.216)
%$$M_t^D = P_t C_t $$
%\item Next two slides explain where money demand comes from and reminds you of some key concepts from EC1B1: ``velocity" and ``quantity equation"
%\ei
%\end{frame}
%
%
%
%\begin{frame}{Recall from EC1B1: ``velocity" and ``quantity equation"}
%\vspace{-2mm}
%\bi
%\item \tr{\textbf{Definition:}} \tr{Velocity $V_t$} is the average number of times a piece of money turns over (i.e. changes hands) in a year
%\bigskip
%\item Consider economy w nominal GDP = total \$ amount of purchases = $P_t Y_t$
%\bigskip
%\item Velocity answers Q: how large is economy's required money stock?
%\bi
%\medskip
%\item If money can be used only once $V_t=1 \Rightarrow M_t^D = P_t Y_t$
%\medskip
%\item If money can be used twice $V_t=2 \Rightarrow M_t^D = P_t Y_t/2$
%\medskip
%\item If money can be used $V_t$ times 
%\begin{equation}\label{eq:quantity}\tag{$\ast$}
%M_t^D V_t = P_t Y_t
%\end{equation}
%\ei
%\item Equation \eqref{eq:quantity} is called \tr{quantity equation}
%\bigskip
%\item Really just an accounting identity, one use: observe nominal GDP $P_t Y_t$ and money stock $M_t$, then velocity $V_t = P_t Y_t/M_t$
%\ei
%\end{frame}
%
%
%\begin{frame}{Different theories of velocity or (equivalently) money demand}
%(Note: 2-4 not examinable, only have to know that there exist different theories)
%\be
%\bigskip
%\item Quantity theory: \tr{$V_t = V$} fixed and hence $M_t \tr{V} = P_t Y_t$
%\bigskip
%\item Other theories of money demand covered in EC1B1, for example the one Lecture 8 in which velocity is given by \tr{$\log V_t = \phi i_t + v_t$}
%\bigskip
%\item Baumol-Tobin model: see supplement
%\bigskip
%\item Cash-in-advance (CIA) models: purchasing goods requires some cash-on-hand which has to be put aside in advance \tr{$P_t C_t \leq M_t$}
%\ee
%\end{frame}
%
%
%\begin{frame}{Mankiw-Weinzierl's version of quantity equation}
%\bi
%\item Follow Mankiw-Weinzierl and assume \tr{quantity theory} with fixed velocity
%\bigskip
%\item But MW write things slightly differently (see p.216)
%$$\mathbb{M}_t = \phi P_t C_t$$
%\item Two differences to standard quantity theory $M_t V = P_t Y_t$
%\be
%\smallskip
%\item MW's quantity equation only applies to purchases of consumption goods (like in CIA model) $\mathbb{M}_t V = P_t C_t$
%\smallskip
%\item MW use different notation $V = \frac{1}{\phi}$
%\ee
%\bigskip
%\item Finally, MW simply redefine $M_t = \mathbb{M}_t V = \mathbb{M}_t/\phi$ and write
%$$\tr{M_t = P_t C_t}$$
%\item Will use this version going forward
%\ei
%\end{frame}


\begin{frame}{Introducing money: money supply and demand}
\bi
\item Price level $P_t$  will end up being determined by monetary policy and equilibrium in the money market
\bi
\medskip
\item money demand = money supply, $M_t^D = M_t^S$
\medskip
\item will sometimes drop $D$ and $S$ superscripts and just write $M_t$
\medskip
\item note: $M_t$ = \tr{stock} of money
\ei
\bigskip
\item Money supply: central bank (monetary policy) sets money supply $M_t^S$
\bigskip
\item Money demand: next slide
\ei
\end{frame}


\begin{frame}{Money demand}
\vspace{-3mm}
\bi
\item Money demand: follow Mankiw-Weinzierl and assume (see p.216)
\begin{equation}\label{eq:MD}\tag{$\ast$}
M_t^D = P_t C_t 
\end{equation}
\item Intuition: need cash to buy goods (``cash-in-advance constraint")
\bigskip
\item Supplement: 
\bi
\medskip
\item flesh out in more detail where \eqref{eq:MD} comes from
\medskip
\item connect to alternative theories of money demand
\medskip
\item revisit some key concepts that should already be familiar from EC1B1, e.g. 
``velocity of money" and ``quantity equation" $M_t V_t = P_t Y_t$
\medskip
\item show that another way of thinking about \eqref{eq:MD} = quantity theory
$$M_t V = P_t Y_t \quad \mbox{with fixed velocity $V$}$$
= root of logic that printing money causes inflation
\ei
\ei
\end{frame}



\begin{frame}{Monetary policy in this model: $i_1$ and $M_2$}
\bi
\item Follow MW and assume central bank has two policy instruments
\be
\medskip
\item $i_1$: nominal interest rate between periods $1$ and $2$
\medskip
\item $M_2^S$: money supply in period $2$
\ee
\bigskip
\item Money supply in period $1$, $M_1^S$, will be whatever is needed to implement $i_1$
\bi
\medskip
\item more on this later (Lecture 8)
\ei
\bigskip
\item Why is monetary policy not only setting $i_1$? Why not only $(M_1^S,M_2^S)$?
\bi
\medskip
\item more on this later (Lecture 8)
\ei
\ei
\end{frame}







\begin{frame}
\renewcommand{\baselinestretch}{1}
\vspace{3cm}
\begin{center}
{\huge Flexible Prices: Monetary Neutrality}
\end{center}
\end{frame}

\begin{frame}{Flexible prices: monetary neutrality}
\vspace{-2mm}
\bi
\item Start by considering version with flexible prices $P_1$ and $P_2$
\bigskip
\item Next: sticky prices = defining assumption of New Keynesian model
\bigskip
\item \tr{\textbf{Definition:}} \tr{Neutrality of money} means that a change in monetary variables like nominal interest rates or the stock of money
\bi
\medskip
\item affects only nominal variables such as prices and nominal wages
\medskip
\item but has no effect on real (inflation-adjusted) variables, like employment, real GDP, and real consumption
\ei
\bigskip
\item Will show: with flexible prices, monetary neutrality holds in our economy
\bi
\medskip
\item real variables $(C_1,C_2,I_1,Y_1,Y_2)$ do not depend on central bank's policy tools, $i_1$ or $M_2$
\medskip
\item only nominal variables, $P_1$ and $P_2$ do
\medskip
\item later: with sticky prices monetary neutrality no longer holds
\ei
\ei
\end{frame}






\begin{frame}{Competitive equilibrium with flexible prices}
\vspace{-3mm}
{\small \tr{\textbf{Definition:}} a competitive equilibrium are quantities $(C_1,C_2,I_1,K_2,Y_1,Y_2,M_1,M_2)$ and prices $(i_1,P_1,P_2)$ such that
\be
\item Utility maximization: taking as given $(i_1,P_1,P_2)$ and $\Omega$, households choose $(C_1,C_2)$ to solve
$$\max_{C_1,C_2} \ U(C_1) + \beta U(C_2) \quad \mbox{s.t.} \quad P_1C_1 + \frac{P_2C_2}{1+i_1} = \Omega$$
where $\Omega$ is the PDV of nominal firm profits (because households own firms)
\item Profit maximization: firms maximize $\Omega = P_1\Pi_1 + \frac{P_2\Pi_2}{1+i_1}$ or equivalently
$$\Omega = \max_{K_2} \ \left\{P_1 (A_1 K_1 - I_1) + \frac{P_2 A_2 K_2}{1+i_1} \right\}, \quad K_2 = I_1, \ Y_1 = A_1 K_1, \ Y_2 = A_2 K_2 $$
\item Market clearing: demand = supply for goods
\begin{align*}
\mbox{goods:} \quad &C_1 + I_1 = Y_1, \qquad C_2 = Y_2\\
\mbox{money:} \quad&P_1 C_1 = M_1, \qquad P_2 C_2 = M_2
\end{align*}
\vspace{-7mm}
\item Policy: $(i_1,M_2)$ are set exogenously by monetary policy 
\ee}
\end{frame}




\begin{frame}{Flexible prices: monetary neutrality}
Proof that flexible prices $\Rightarrow$ monetary neutrality is simple and general
\\
\bigskip
This slide: steps of general proof that works in large class of models\\
\bigskip
Next slides: apply steps to particular 2-period model we're covering\\
\bigskip
Steps of general proof: with flexible prices
\be
\medskip
\item can rewrite all equations in real terms by dividing through by price level
\medskip
\item this results in a block of self-contained equilibrium conditions for the real variables that \tr{do not depend on any nominal variables}
\medskip
\item therefore the equilibrium real variables \tr{do not depend on nominal variables} $\Rightarrow$ monetary neutrality
\ee
\end{frame}


\begin{frame}{Flexible prices: monetary neutrality}
Application to our particular 2-period model: with flexible prices
\be
\medskip
\item can rewrite all equations in real terms by dividing through by price level
\bi
\smallskip
\item divide through by $P_1$
\ei
\medskip
\item this results in a block of self-contained equilibrium conditions for the real variables that \tr{do not depend on any nominal variables}
\bi
\smallskip
\item this block = same as equilibrium conditions in ``baby RBC model"
\ei
\medskip
\item therefore the equilibrium real variables \tr{do not depend on nominal variables} $\Rightarrow$ monetary neutrality
\bi
\smallskip
\item real allocation same as in baby RBC model
\ei
\ee
\end{frame}


\begin{frame}{Example of step 1: rewrite budget constraint in real terms}
\bi
\item Recall household budget constraint in nominal terms
$$P_1 C_1 + \frac{P_2 C_2}{1+i_1} = P_1\Pi_1 + \frac{P_2\Pi_2}{1+i_1}$$
\item Divide by $P_1$
$$C_1 + \frac{P_2 C_2}{P_1(1+i_1)} = \Pi_1 + \frac{P_2\Pi_2}{P_1(1+i_1)}$$
\item Use definition of real interest rate $1+r_1 = (1+i_1)P_1/P_2$
$$C_1 + \frac{C_2}{1+r_1} = \Pi_1 + \frac{\Pi_2}{1+r_1}$$
\item Therefore clearly, solution to household's problem is same as before
\bigskip
\item Next follow same logic for entire economy
\ei
\end{frame}



\begin{frame}{Step 1: rewrite all equations in real terms ... }
\vspace{-3mm}
{\small \tr{\textbf{Definition:}} a competitive equilibrium are quantities $(C_1,C_2,I_1,K_2,Y_1,Y_2,M_1,M_2)$ and prices $(i_1,P_1,P_2)$ such that
\be
\item Utility maximization: taking as given $(i_1,P_1,P_2)$ and $\Omega$, households choose $(C_1,C_2)$ to solve
$$\max_{C_1,C_2} \ U(C_1) + \beta U(C_2) \quad \mbox{s.t.} \quad P_1C_1 + \frac{P_2C_2}{1+i_1} = \Omega$$
where $\Omega$ is the PDV of nominal firm profits (because households own firms)
\item Profit maximization: firms maximize $\Omega = P_1\Pi_1 + \frac{P_2\Pi_2}{1+i_1}$ or equivalently
$$\Omega = \max_{K_2} \ \left\{P_1 (A_1 K_1 - I_1) + \frac{P_2 A_2 K_2}{1+i_1} \right\}, \quad K_2 = I_1, \ Y_1 = A_1 K_1, \ Y_2 = A_2 K_2 $$
\item Market clearing: demand = supply for goods
\begin{align*}
\mbox{goods:} \quad &C_1 + I_1 = Y_1, \qquad C_2 = Y_2\\
\mbox{money:} \quad&P_1 C_1 = M_1, \qquad P_2 C_2 = M_2
\end{align*}
\vspace{-7mm}
\item Policy: $(i_1,M_2)$ are set exogenously by monetary policy 
\ee}
\end{frame}

\begin{frame}{Step 1: ... divide by $P_1$ }
\vspace{-3mm}
{\small \tr{\textbf{Definition:}} a competitive equilibrium are quantities $(C_1,C_2,I_1,K_2,Y_1,Y_2,M_1,M_2)$ and prices $(i_1,P_1,P_2)$ such that
\be
\item Utility maximization: taking as given $(i_1,P_1,P_2)$ and $\Omega$, households choose $(C_1,C_2)$ to solve
$$\max_{C_1,C_2} \ U(C_1) + \beta U(C_2) \quad \mbox{s.t.} \quad C_1 + \frac{P_2C_2}{P_1(1+i_1)} = \frac{\Omega}{P_1}$$
where $\Omega$ is the PDV of nominal firm profits (because households own firms)
\item Profit maximization: firms maximize $\Omega/P_1 = \Pi_1 + \frac{P_2\Pi_2}{P_1(1+i_1)}$ or equivalently
$$\frac{\Omega}{P_1} = \max_{K_2} \ \left\{A_1 K_1 - I_1 + \frac{P_2 A_2 K_2}{P_1(1+i_1)} \right\}, \quad K_2 = I_1, \ Y_1 = A_1 K_1, \ Y_2 = A_2 K_2 $$
\item Market clearing: demand = supply for goods
\begin{align*}
\mbox{goods:} \quad &C_1 + I_1 = Y_1, \qquad C_2 = Y_2\\
\mbox{money:} \quad&P_1 C_1 = M_1, \qquad P_2 C_2 = M_2
\end{align*}
\vspace{-7mm}
\item Policy: $(i_1,M_2)$ are set exogenously by monetary policy 
\ee}
\end{frame}


\begin{frame}{{\large Step 1: ... and use $1+r_1 = (1+i_1)P_1/P_2$ and real firm value $W = \Omega/P_1$}}
\vspace{-3mm}
{\small \tr{\textbf{Definition:}} a competitive equilibrium are quantities $(C_1,C_2,I_1,K_2,Y_1,Y_2,M_1,M_2)$ and prices $(i_1,P_1,P_2)$ such that
\be
\item Utility maximization: taking as given $1+r_1 = (1+i_1)P_1/P_2$ and $W$, households choose $(C_1,C_2)$ to solve
$$\max_{C_1,C_2} \ U(C_1) + \beta U(C_2) \quad \mbox{s.t.} \quad C_1 + \frac{C_2}{1+r_1} = W$$
where $W$ is the PDV of real firm profits (because households own firms)
\item Profit maximization: firms maximize $W = \Pi_1 + \frac{\Pi_2}{1+r_1}$ or equivalently
$$W = \max_{K_2} \ \left\{A_1 K_1 - I_1 + \frac{A_2 K_2}{1+r_1} \right\}, \quad K_2 = I_1, \ Y_1 = A_1 K_1, \ Y_2 = A_2 K_2 $$
\item Market clearing: demand = supply for goods
\begin{align*}
\mbox{goods:} \quad &C_1 + I_1 = Y_1, \qquad C_2 = Y_2\\
\mbox{money:} \quad&P_1 C_1 = M_1, \qquad P_2 C_2 = M_2
\end{align*}
\vspace{-7mm}
\item Policy: $(i_1,M_2)$ are set exogenously by monetary policy 
\ee}
\end{frame}


\begin{frame}{{\large Step 2: real variables solve eqn's that do not depend on nominal stuff}}
\vspace{-3mm}
{\small \tr{Real quantities $(C_1,C_2,I_1,K_2,Y_1,Y_2)$ and the price $r_1$} are such that
\be
\item Utility maximization: taking as given $r_1,W$, households choose $(C_1,C_2)$ to solve
$$\max_{C_1,C_2} \ U(C_1) + \beta U(C_2) \quad \mbox{s.t.} \quad C_1 + \frac{C_2}{1+r_1} = W$$
where $W$ is the PDV of real firm profits (because households own firms)
\item Profit maximization: firms maximize $W = \Pi_1 + \frac{\Pi_2}{1+r_1}$ or equivalently
$$W = \max_{K_2} \ \left\{A_1 K_1 - I_1 + \frac{A_2 K_2}{1+r_1} \right\}, \quad K_2 = I_1, \ Y_1 = A_1 K_1, \ Y_2 = A_2 K_2 $$
\item Market clearing: $C_1 + I_1 = Y_1$ and $C_2 = Y_2$
\ee
\medskip
\tb{Nominal quantities $(M_1,M_2)$ and prices $(i_1,P_1,P_2)$} are such that
\be
\item Money market: $P_1 C_1 = M_1$ and $P_2 C_2 = M_2$
\item Policy: $(i_1,M_2)$ are set exogenously by monetary policy 
\item Definition of real interest rate: $(1+i_1)P_1/P_2 = 1+r_1$
\ee}
\end{frame}



\begin{frame}{Step 3: therefore solution does not depend on nominal stuff}
\vspace{-3mm}
{\small \tr{Equilibrium real quantities $(C_1,C_2,I_1,K_2,Y_1,Y_2)$ and the price $r_1$} are given by
\begin{equation*}\begin{split}
C_1 &= \frac{\left(\frac{1}{\beta A_2} \right)^\sigma A_2}{1+\left(\frac{1}{\beta A_2} \right)^\sigma A_2}A_1 K_1 \\
C_2 &= \frac{A_2}{1+\left(\frac{1}{\beta A_2} \right)^\sigma A_2}A_1 K_1 \\
K_2 = I_1 &= \frac{1}{1+\left(\frac{1}{\beta A_2} \right)^\sigma A_2}A_1 K_1 \\
Y_1 &= A_1 K_1 \\
Y_2 &= \frac{A_2}{1+\left(\frac{1}{\beta A_2} \right)^\sigma A_2}A_1 K_1\\
1+r_1&=A_2
\end{split}
\end{equation*}
\tr{Clearly solution does not depend on nominal variables $\Rightarrow$ monetary neutrality!}
}
\end{frame}



\begin{frame}{What about the nominal variables?}
\vspace{-3mm}
Recall: \tb{Nominal quantities $(M_1,M_2)$ and prices $(i_1,P_1,P_2)$} are such that
\be
\smallskip
\item Money market: $P_1 \tr{C_1} = M_1$ and $P_2 \tr{C_2} = M_2$
\smallskip
\item Policy: $(i_1,M_2)$ are set exogenously by monetary policy 
\smallskip
\item Definition of real interest rate: $(1+i_1)P_1/P_2 = 1+\tr{r_1}$
\ee
\medskip
Already know \tr{$(C_1,C_2,r_1)$} so have 3 equations for 3 unknowns \tb{$(P_1,P_2,M_1)$}\\
\medskip
Can show (see supplement):
\begin{equation*}
\begin{split}
\tb{P_1} &= \frac{1+\left(\frac{1}{\beta A_2} \right)^\sigma A_2}{A_1 K_1} \tb{\frac{M_2}{1+i_1}} \\
\tb{P_2} &= \frac{1+\left(\frac{1}{\beta A_2} \right)^\sigma A_2}{A_2A_1 K_1} \tb{M_2} \\
\tb{M_1} &= \left(\frac{1}{\beta A_2} \right)^\sigma A_2 \tb{\frac{M_2}{1+i_1}}
\end{split}
\end{equation*}
\end{frame}



\begin{frame}{Classical dichotomy}
\bi
\item \tr{\textbf{Definition:}} an economy displays the \tr{classical dichotomy} if all real variables are determined independently of the nominal variables and therefore real and nominal variables can be analyzed separately
\bigskip
\item Clearly the model with flexible prices has this feature
\bigskip
\item Also see wikipedia: {\scriptsize \url{https://en.wikipedia.org/wiki/Classical_dichotomy}}
\ei
\end{frame}



\begin{frame}{Classical dichotomy can be seen clearly in this slide}
\vspace{-3mm}
{\small \tr{Real quantities $(C_1,C_2,I_1,K_2,Y_1,Y_2)$ and the price $r_1$} are such that
\be
\item Utility maximization: taking as given $r_1,W$, households choose $(C_1,C_2)$ to solve
$$\max_{C_1,C_2} \ U(C_1) + \beta U(C_2) \quad \mbox{s.t.} \quad C_1 + \frac{C_2}{1+r_1} = W$$
where $W$ is the PDV of real firm profits (because households own firms)
\item Profit maximization: firms maximize $W = \Pi_1 + \frac{\Pi_2}{1+r_1}$ or equivalently
$$W = \max_{K_2} \ \left\{A_1 K_1 - I_1 + \frac{A_2 K_2}{1+r_1} \right\}, \quad K_2 = I_1, \ Y_1 = A_1 K_1, \ Y_2 = A_2 K_2 $$
\item Market clearing: $C_1 + I_1 = Y_1$ and $C_2 = Y_2$
\ee
\medskip
\tb{Nominal quantities $(M_1,M_2)$ and prices $(i_1,P_1,P_2)$} are such that
\be
\item Money market: $P_1 C_1 = M_1$ and $P_2 C_2 = M_2$
\item Policy: $(i_1,M_2)$ are set exogenously by monetary policy 
\item Definition of real interest rate: $(1+i_1)P_1/P_2 = 1+r_1$
\ee}
\end{frame}


\begin{frame}{Monetary policy only affects price level and inflation}
\vspace{-3mm}
\bi
\item Recall monetary policy instruments $(i_1,M_2)$ 
\medskip
\item What if central bank cuts interest rate $i_1$ or prints more money $M_2$?
\medskip
\item Already know: monetary policy has no effect on real variables. Does it affect anything?
\medskip
\item Yes: price level and inflation. Recall:
\smallskip
\ei
{\small $$\tb{P_1} = \frac{1+\left(\frac{1}{\beta A_2} \right)^\sigma A_2}{A_1 K_1} \tb{\frac{M_2}{1+i_1}}, \quad \tb{P_2} = \frac{1+\left(\frac{1}{\beta A_2} \right)^\sigma A_2}{A_2A_1 K_1} \tb{M_2}, \quad \tb{1+\pi_2 = \frac{P_2}{P_1}} = \frac{1 + \tb{i_1}}{A_2}$$}
\vspace{-4mm}
\bi
\medskip
\item Printing money $M_2$ increases price level $P_1,P_2$ (classic quantity theory)
\bi
\smallskip
\item can also show: higher money growth $M_2/M_1 \Rightarrow$ higher inflation $\pi_2$
\ei
\medskip
\item Interest rate cut $i_1 \downarrow$
\bi
\smallskip
\item raises price level $P_1$ and short-run inflation, $1+\pi_1 = P_1/P_0$
\smallskip
\item counterintuitively, \tr{lowers} long-run inflation $\pi_2$, i.e. interest rate cuts are deflationary in long-run (``Neo-Fisherism")
\ei
\ei
\end{frame}



\begin{frame}{Alternative way of thinking about determination of $P_1$: aggregate demand = aggregate supply (only useful later)}
\vspace{-3mm}
\bi
\item Recall goods market clearing condition in period 1
$$C_1 + I_1 = Y_1 \quad \mbox{with} \quad Y_1 = A_1 K_1 $$
\item Can think about left-hand side as aggregate demand
$$Y_1^D = C_1 + I_1 $$
and right-hand side as aggregate supply
$$Y_1^S = A_1 K_1 $$
\item By doing some algebra, can write $Y_1^D$ as function of price level $P_1$
\bigskip
\item Note: this mixes real and nominal variables
\bi
\smallskip
\item given classical dichotomy, a very strange thing to do
\smallskip
\item but let's do it anyway because it will be useful later when classical dichotomy fails with sticky prices
\ei
\ei
\end{frame}




\begin{frame}{Aggregate demand = aggregate supply (only useful later)}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.6\textwidth]{figures/lec7_1.eps} 
\end{figure}
\vspace{-5mm}
Can check: price level that equates aggregate demand and supply is
$$P_1^* = \frac{1+\left(\frac{1}{\beta A_2} \right)^\sigma A_2}{A_1 K_1} \frac{M_2}{1+i_1}$$
\end{frame}



\begin{frame}{Monetary policy only affects price level and inflation}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.6\textwidth]{figures/lec7_3.eps} 
\end{figure}
\vspace{-5mm}
Monetary policy \tb{$i_1 \downarrow$} or \tb{$M_2 \uparrow$} increases $Y_1^D$ but this only shows up in $P_1 \uparrow$
$$Y_1^D = \left[1 + \left(\frac{1}{\beta A_2} \right)^\sigma A_2 \right] \frac{\tb{M_2}}{\tb{(1+i_1)}P_1}$$
\end{frame}







\begin{frame}{Flexible prices: summary and policy implications}
\vspace{-3mm}
\bi
\item Real variables are completely separate from nominal variables\\
(``monetary neutrality", ``classical dichotomy")
\bigskip
\item Corollary: monetary policy has no effect on any real variables
\bigskip
\item Monetary policy affects only price level and inflation
\bigskip
\item 1st welfare theorem still holds: policy intervention undesirable in first place
\bigskip
\item \tr{Stabilization policy?} When there is a recession (e.g. due to $A_1$ or $A_2 \downarrow$)
\bi
\medskip
\item stabilization via fiscal policy is undesirable
\medskip
\item stabilization via monetary policy is not possible (even if it were, it would be undesirable)
\ei
\medskip
\item \tr{Role of central bank?} It can manage price level and inflation but those do not matter so may as well close down central bank
\ei
\end{frame}



\begin{frame}{{\large Only one problem: empirical evidence rejects monetary neutrality}}
\vspace{-4mm}
\bi
\item For summary of evidence, see section ``Monetary Policy: What Is the Best Evidence We Have?" in Nakamura and Steinsson (2018)
\medskip
\item One test of neutrality: do \tr{real} interest rates change when the central bank changes \tr{nominal} interest rates? If they do, so will other real variables.
\bi
\smallskip
\item monetary neutrality: they don't, only inflation responds
\smallskip
\item in our model: $1+r_1 = A_2$ regardless of $i_1$
\ei
\smallskip
\item A number of papers study real rates and find non-neutrality. Example:
\begin{figure}[tph]
	\centering
\includegraphics[width=.6\textwidth]{figures/NS_abstract} 
\end{figure}
\ei
\end{frame}


\begin{frame}
\renewcommand{\baselinestretch}{1}
\vspace{3cm}
\begin{center}
{\huge Sticky Prices: Monetary Non-Neutrality}
\end{center}
\end{frame}

\begin{frame}{Sticky prices}
\bi
\item Will change one single assumption
\bi
\medskip
\item that prices are perfectly flexible in the short run (period 1)
\ei
\bigskip
\item This will change results and policy implications dramatically
\bigskip
\item Recall: first welfare theorem holds when there are \tr{no ``frictions"}
\bigskip
\item \tr{Sticky prices are exactly such a friction} $\Rightarrow$ break first welfare theorem
\bigskip
\item Alternative assumption with similar implications: \tr{sticky wages}
\bigskip
\item Before developing model: why do we think prices could be sticky?
\ei
\end{frame}


\begin{frame}{Reasons for price stickiness}
Many different theories of sticky prices. Two most common ones:
\be
\bigskip
\item Menu costs:
\bi
\medskip
\item firm pays fixed cost to change prices (e.g. print new restaurant menu)
\medskip
\item $\Rightarrow$ only change price when payoff is large enough to cover fixed cost
\ei
\bigskip
\item (Rational) inattention:
\bi
\medskip
\item acquiring information is costly
\medskip
\item firms optimally choose to not  pay attention to what's going on all the time, in particular monetary policy
\ei
\ee
\end{frame}





\begin{frame}{Empirical evidence on price stickiness}
\begin{figure}[tph]
	\centering
\includegraphics[width=.6\textwidth]{figures/goldberg-hellerstein-abstract}\includegraphics[width=.4\textwidth]{figures/goldberg-hellerstein-fig} 
\end{figure}
Note: Britannia = fictional name of ``popular British beer brand"\\
\bigskip
{\notsotiny Source: Goldberg and Hellerstein (2007) ``Sticky Prices: Why Firms Hesitate to Adjust the Price of Their Goods" \url{https://www.newyorkfed.org/medialibrary/media/research/current_issues/ci13-10.pdf} }
\end{frame}


\begin{frame}{Empirical evidence on wage stickiness}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.77\textwidth]{figures/hazell} 
\end{figure}
Source: Hazell (2022) ``Downward Rigidity in the Wage for New Hires"
\end{frame}



\begin{frame}{Modeling price stickiness in practice}
\vspace{-2mm}
\bi
\item Menu costs, inattention are simple ideas but surprisingly hard to model
\bi
\medskip
\item models usually too complicated to embed in full macro model
\ei
\bigskip
\item In practice often Calvo pricing: firms are allowed to change price with exogenous probability $\alpha$ (``Calvo fairy")
\bigskip
\item Calvo pricing is example of \tr{time-dependent} sticky prices
\bigskip
\item In contrast to \tr{state-dependent} models, e.g. menu costs, inattention
\bigskip
\item State-depedent models often considered more satisfactory
\bigskip
\item Paul Krugman in characteristically provocative fashion:\\ ``While I regard the evidence for such stickiness as overwhelming, the assumption of at least temporarily rigid nominal prices is one of those things that works beautifully in practice but very badly in theory." ({\scriptsize \url{https://web.mit.edu/krugman/www/islm.html}})
\ei
\end{frame}



\begin{frame}{Our assumption: $P_1$ is completely fixed}
\bi
\item Will make even more simplistic assumption: $P_1$ is \tr{completely fixed}
\bi
\medskip
\item idea: firms have set prices some time before period 1, before knowing economic conditions like $A_1,A_2$ etc, can not change them anymore
\medskip
\item so $P_1 = P_0$ where $P_0$ is reference price set in pre-period 0 from earlier
\ei
\bigskip
\item In contrast $P_2$ is flexible
\bigskip
\item Captures common idea in macroeconomics: prices are sticky in short run (period 1) but flexible in long run (period 2)
\bigskip
\item End of slides: briefly discuss less extreme assumption = partial stickiness
\ei
\end{frame}



\begin{frame}{Preview: equilibrium with sticky prices}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.8\textwidth]{figures/lec7_1.eps} 
\end{figure}
\end{frame}



\begin{frame}{{\large Preview: with sticky prices, a negative demand shock causes recession}}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.8\textwidth]{figures/lec7_2.eps} 
\end{figure}
\end{frame}


\begin{frame}{Preview: monetary or fiscal policy can counteract recession}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.8\textwidth]{figures/lec7_5.eps} 
\end{figure}
\end{frame}


\begin{frame}{What is a negative demand shock?}
In our simple model, two parameter capture demand effects
\be
\bigskip
\item A change in beliefs at $t=1$ that \tr{future productivity $A_2 \downarrow$} so that households suddenly become more pessimistic about the future
\bi
\medskip
\item called a ``news shock" in economics literature
\medskip
\item drop in $A_2$ does not even have to materialize, it's enough that households believe it will happen
\medskip
\item if drop in $A_2$ materializes, $A_2 \downarrow$ is also a supply shock at $t=2$
\ei
\bigskip
\item \tr{Patience $\beta \uparrow$} so that households suddenly become more thrifty, want to spend less and save more
\ee
\bigskip
Will see clearly in equations below why these two parameters capture demand effects
\end{frame}




\begin{frame}{Equilibrium with sticky prices}
\vspace{-3mm}
{\small \tr{\textbf{Def'n:}} an equilibrium are quantities $(C_1,C_2,I_1,K_2,Y_1,Y_2,M_1,M_2)$ \& prices $(i_1,P_1,P_2)$
\be
\item Utility maximization: taking as given $(i_1,P_1,P_2)$ and $\Omega$, households choose $(C_1,C_2)$ to solve
$$\max_{C_1,C_2} \ U(C_1) + \beta U(C_2) \quad \mbox{s.t.} \quad P_1C_1 + \frac{P_2C_2}{1+i_1} = \Omega$$
where $\Omega$ is the PDV of nominal firm profits (because households own firms)
\item Profit maximization: firms maximize $\Omega = P_1\Pi_1 + \frac{P_2\Pi_2}{1+i_1}$ or equivalently
$$\Omega = \max_{K_2} \ \left\{P_1 (\tr{Y_1} - I_1) + \frac{P_2 A_2 K_2}{1+i_1} \right\}, \quad K_2 = I_1, \ \tr{Y_1^S = A_1 K_1}, \ Y_2 = A_2 K_2 $$
\item Market clearing: demand = supply for goods
\begin{align*}
\mbox{goods:} \quad &\tr{Y_1 = \min\{Y_1^D,Y_1^S\} \ \mbox{ with } \ Y_1^D = C_1 + I_1}, \qquad C_2 = Y_2\\
\mbox{money:} \quad&P_1 C_1 = M_1, \qquad P_2 C_2 = M_2
\end{align*}
\vspace{-7mm}
\item Policy: $(i_1,M_2)$ are set exogenously by monetary policy \\[-0.5ex]
\item \tr{Sticky prices: $P_1$ is completely fixed at some exogenous value}
\ee}
\end{frame}







\begin{frame}{Explanation of new terms on previous slide}
\vspace{-3mm}
\bi
\item Perfectly sticky $P_1 \Rightarrow$ goods market at $t=1$ not necessarily in equilibrium
\bi
\medskip
\item see graph on earlier slide
\medskip
\item recall: agg demand $= Y_1^D = C_1 + I_1$, agg supply $=Y_1^S = A_1 K_1$
\ei
\bigskip
\item Actual GDP $Y_1$ is determined by \tr{short side of the market}
\bi
\medskip
\item when demand is strong, GDP is at potential: $Y_1 = Y_1^S = A_1 K_1$
\medskip
\item when demand is weak, GDP is below potential: $Y_1 = Y_1^D < A_1 K_1$
\medskip
\item more precisely $\tr{Y_1 = \min\{Y_1^D,Y_1^S\}} = \min\{C_1 + I_1,A_1 K_1\}$
\ei
\bigskip
\item Will sometimes call $Y_1^S = A_1 K_1 = $ \tr{``potential GDP"} and $Y_1 =$ \tr{``actual GDP"} and gap between the two $Y_1 - Y_1^S =$ \tr{``output gap"}
\medskip
\item Interesting case: a negative demand shock pushes GDP below potential
\medskip
\item \textbf{\tr{Assumption:}} parameters are such that $Y_1^D \leq Y_1^S$, i.e. demand is either at potential or below (never have excess demand, $Y_1^D > Y_1^S$)
\ei
\end{frame}






\begin{frame}{{\large Equilibrium with sticky prices (also see Mankiw-Weinzierl's equations)}}
\vspace{-2mm}
{\small The equilibrium allocation with \tr{perfectly sticky $P_1$} equals
\begin{equation*}\begin{split}
C_1 &= \left(\frac{1}{\beta A_2} \right)^\sigma A_2 \frac{M_2}{(1+i_1)P_1} \\
C_2 &= A_2 \frac{M_2}{(1+i_1)P_1}\\
I_1 &= \frac{M_2}{(1+i_1)P_1} \\
Y_1 &= \left[1 + \left(\frac{1}{\beta A_2} \right)^\sigma A_2 \right] \frac{M_2}{(1+i_1)P_1}\\
Y_2 &= A_2 \frac{M_2}{(1+i_1)P_1}\\
P_2 &= \frac{1+i_1}{A_2}P_1
\end{split}\end{equation*}
Note: equations are identical to \tr{MW's equations (31)-(36) with $G_1=0$ and $g_2=0$}}
\end{frame}


\begin{frame}{Sticky prices break monetary neutrality}
\vspace{-2mm}
Clearly allocation now depends on \tb{nominal variables} $\Rightarrow$ \tr{\sout{monetary neutrality}!}\\
\begin{equation*}\begin{split}
C_1 &= \left(\frac{1}{\beta A_2} \right)^\sigma A_2 \tb{\frac{M_2}{(1+i_1)P_1}} \\
C_2 &= A_2 \tb{\frac{M_2}{(1+i_1)P_1}}\\
I_1 &= \tb{\frac{M_2}{(1+i_1)P_1}} \\
Y_1 &= \left[1 + \left(\frac{1}{\beta A_2} \right)^\sigma A_2 \right] \tb{\frac{M_2}{(1+i_1)P_1}}\\
Y_2 &= A_2 \tb{\frac{M_2}{(1+i_1)P_1}}\\
P_2 &= \frac{1+i_1}{A_2}P_1
\end{split}\end{equation*}
\end{frame}



\begin{frame}{{\large With sticky prices, a negative demand shock causes a recession}}
\vspace{-4mm}
Recall: two parameter capture demand effects
\be
\medskip
\item A change in beliefs at $t=1$ that \tr{$A_2 \downarrow$} so that households suddenly become more pessimistic about the future
\medskip
\item Patience \tr{$\beta \uparrow$} so that households want to spend less and save more
\ee
\medskip
\tr{These cause recession at $t=1$} (recall assumption that $\sigma<1$ -- see lecture 5):
\only<all:1>{
\medskip
\begin{equation*}\begin{split}
C_1 &= \tr{\left(\frac{1}{\beta A_2} \right)^\sigma A_2} \frac{M_2}{(1+i_1)P_1} \\[1ex]
Y_1 &= \left[1 + \tr{\left(\frac{1}{\beta A_2} \right)^\sigma A_2} \right] \frac{M_2}{(1+i_1)P_1}\\
\end{split}\end{equation*}
In contrast, \tr{with flexible prices, GDP $Y_1 = A_1 K_1$ is unaffected} when $A_2 \downarrow$ or $\beta \uparrow$
}
\only<all:2>{
\vspace{-6mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.5\textwidth]{figures/lec7_2.eps} 
\end{figure}
}
\end{frame}



\begin{frame}{Key observation: with sticky prices household income depends on aggregate demand}
\vspace{-4mm}
Recall household budget constraint and firm's problem
$$P_1C_1 + \frac{P_2C_2}{1+i_1} = \Omega, \qquad \Omega = \max_{K_2} \ \left\{P_1 (\tr{Y_1} - I_1) + \frac{P_2 A_2 K_2}{1+i_1} \right\}, \quad \tr{Y_1 = Y_1^D}$$
Using optimality condition for investment: $P_1 = P_2 A_2/(1+i_1)$
$$P_1C_1 + \frac{P_2C_2}{1+i_1} = \Omega, \qquad \Omega = P_1 \tr{Y_1}, \qquad \tr{Y_1 = Y_1^D}$$
Intuition: \textbf{\tr{one person's spending is someone else's income}}\\
\smallskip
Generates \tr{Keynesian-cross-type feedback loop}:
\bi
\smallskip
\item  $C_1 \downarrow \Rightarrow Y_1^D = C_1 + I_1 \downarrow \Rightarrow Y_1 = Y_1^D \downarrow \Rightarrow \Omega = P_1 Y_1 \downarrow \Rightarrow C_1 \downarrow$ and so on
\smallskip
\item Here this works through dividend payments: demand $\downarrow$ $\Rightarrow$ firm production $\downarrow$ $\Rightarrow$ firm profits and dividend payments $\downarrow$ $\Rightarrow$ household capital income $\downarrow$
\smallskip
\item In more general models also through labor income: demand $\downarrow$ $\Rightarrow$ firm production $\downarrow$ $\Rightarrow$ household labor income $\downarrow$
\ei
\end{frame}



\begin{frame}{More on $\beta \uparrow$: the ``Paradox of Thrift"}
\vspace{-2mm}
Compare effect of $\beta \uparrow$ = want to spend less, save more
\bi
\medskip
\item \tr{\textbf{Flexible prices:}} \only<all:2-5>{aggregate saving and investment increase as households cut current consumption in return for future consumption}
\only<all:1>{
\begin{equation*}\begin{split}
C_1 &= \frac{\left(\frac{1}{\tr{\beta} A_2} \right)^\sigma A_2}{1+\left(\frac{1}{\tr{\beta} A_2} \right)^\sigma A_2}A_1 K_1 \quad \tr{\downarrow} \\
I_1 &= \frac{1}{1+\left(\frac{1}{\tr{\beta} A_2} \right)^\sigma A_2}A_1 K_1 \quad \tr{\uparrow} \\
Y_1 &= A_1 K_1 \quad \tr{\mbox{unaffected}} \\
C_2 = Y_2 &= \frac{A_2}{1+\left(\frac{1}{\tr{\beta} A_2} \right)^\sigma A_2}A_1 K_1 \quad \tr{\uparrow}\\
\end{split}
\end{equation*}}
\only<all:2-4>{
$$Y_1 \ \mbox{unaffected}, \quad C_1 \downarrow, \quad I_1, \ C_2, \ Y_2 \ \mbox{all} \uparrow $$
}
\item \tr{\textbf{Sticky prices:}} \only<all:4-5>{even though everyone wants to save more, aggregate saving and investment do not increase = \tr{``paradox of thrift"}}
\only<all:3>{
\begin{equation*}\begin{split}
C_1 &= \left(\frac{1}{\tr{\beta} A_2} \right)^\sigma A_2 \frac{M_2}{(1+i_1)P_1} \quad \tr{\downarrow} \\
I_1 &= \frac{M_2}{(1+i_1)P_1} \quad \tr{\mbox{unaffected}} \\
Y_1 &= \left[1 + \left(\frac{1}{\tr{\beta} A_2} \right)^\sigma A_2 \right] \frac{M_2}{(1+i_1)P_1} \quad \tr{\downarrow}\\
C_2 = Y_2 &= A_2 \frac{M_2}{(1+i_1)P_1} \quad \tr{\mbox{unaffected}}
\end{split}\end{equation*}
}
\only<all:4-5>{
$$C_1 \downarrow, \quad Y_1 \downarrow, \quad I_1, \ C_2, \ Y_2 \ \mbox{all unaffected} $$
}
\only<all:5>{
Intuition in Mankiw-Weinzierl's words:

\bi
{\small\smallskip
\item ``If $\beta$  rises, households will want to consume less and save more"
\smallskip
\item ``In equilibrium, however, saving and investment are unchanged, because output falls"
\smallskip
\item ``That is, because aggregate demand influences output, more thriftiness does not increase equilibrium saving"}
\ei
}
\ei
\end{frame}


\begin{frame}{Paradox of thrift: more detailed intuition}
\vspace{-2mm}
Recall: $\beta \uparrow$ = want to spend less, save more and
$$\beta \uparrow \quad \Rightarrow \quad C_1 \downarrow, \quad Y_1 \downarrow, \quad I_1, \ C_2, \ Y_2 \ \mbox{all unaffected}  $$
What's going on under the hood, i.e. what's transmission mechanism?
\bi
\medskip
\item $\beta \uparrow \Rightarrow$ household consumption $C_1 \downarrow,C_2 \uparrow$ from \tr{Euler eqn} $C_1 = C_2 [\beta(1+r_1)]^{-\sigma}$ (intertemporal substitution)
\medskip
\item also $\beta \uparrow \Rightarrow$ household saving $\uparrow \Rightarrow \tr{I_1 \uparrow}$
\medskip
\item $C_1 \downarrow \Rightarrow$ aggregate demand $Y_1^D = C_1 + I_1 \downarrow$
\medskip
\item $Y_1^D \downarrow \Rightarrow$ output $Y_1 \downarrow \Rightarrow$ household income $\Omega = P_1 Y_1 \downarrow \Rightarrow C_1\downarrow$ (income effect) $\Rightarrow$ aggregate demand $Y_1^D \downarrow \Rightarrow ...$ (Keynesian cross logic)
\medskip
\item also household income $\Omega = P_1 Y_1 \downarrow \Rightarrow$ household saving $\downarrow \Rightarrow$ \tr{$I_1 \downarrow$}
\ei
\bigskip
In equilibrium, the fall in household income is such that \tr{$I_1 \uparrow$} and \tr{$I_1 \downarrow$} above exactly offset and $I_1$ is unaffected. Therefore so are $C_2 = Y_2 = A_2 K_2$.
\end{frame}




\begin{frame}{{\large Comment: equilibrium real interest rate in MW version of NK model}}
\vspace{-4mm}
\bi
\item Odd model property: \tr{real} interest rate is \tr{$1+r_1 = A_2$} regardless of \tb{$i_1,M_2$}
\bi
\smallskip
\item see this from $P_2 = \frac{1+i_1}{A_2}P_1$ and definition $1+r_1 = (1+i_1)P_1/P_2$
\smallskip
\item in equilibrium, \tr{no passthrough} from \tb{nominal} to \tr{real} interest rate
\smallskip
\item surprising given we just made big deal out of monetary \tr{non}-neutrality
\ei
\medskip
\item What's going on? Answer: just an \tr{artefact} of particular model assumption
\bi
\smallskip
\item linear production $Y_2 = A_2 K_2 \Rightarrow$ \tr{infinitely elastic capital demand} \\ $\Rightarrow$ equilibrium real interest rate is $1+r_1 = A_2$ always
\smallskip
\item an assumption we made only for tractability
\smallskip
\item there definitely is passthrough from $i_1$ to $r_1$ and money is non-neutral (whole issue is that it's ``too non-neutral" because of $\infty$ elasticity)
\ei
\medskip
\item Supplement: relax this assumption  $Y_2 = F(K_2) = A_2 K_2^{\tr{\alpha}},0<\tr{\alpha} \leq 1$
$$\tr{1 + r_1} = \alpha^{\alpha} A_2  \left(\tb{\frac{(1+i_1)P_1}{M_2}}\right)^{1-\alpha} \quad \mbox{so that \tr{$r_1$} depends on \tb{$i_1$} as expected}$$
\ei
\end{frame}


\begin{frame}{New Keynesian model and inflation (partial price stickiness)}
\vspace{-3mm}
\bi
\item So far: $P_1$ completely fixed $=P_0$
\bi
\medskip
\item $\Rightarrow$ short-run inflation $\pi_1 = (P_1-P_0)/P_0 = 0$ by assumption
\ei
\bigskip
\item Contrasts with common intuition and prediction from IS-MP-PC model: aggregate demand $\uparrow$ $\Rightarrow$ short-run inflation $\uparrow$
\bigskip
\item Standard New Keynesian model (the one in textbooks e.g. Gali) features partial price stickiness, typically Calvo pricing
\bi
\pause
\medskip
\item need firm price-setting power ($\neq$ competitive equilibrium)
\medskip
\item see Kurlat chapters 14.2 and 14.3 for good discussion
\ei
\bigskip
\item In standard NK model, aggregate demand $\uparrow$ $\Rightarrow$ short-run inflation $\uparrow$
\bi
\medskip
\item one way of thinking about this: aggregate demand/supply diagram with prices adjusting partially
\ei
\ei
\end{frame}


\begin{frame}{{\large Various demand shocks in a fully-fledged New Keynesian model}}
\vspace{-4mm}
{\small Note: figure shows positive demand shocks, not negative ones like previous slides}
\vspace{-2mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.58\textwidth]{figures/smets-wouters-demand} 
\end{figure}
\vspace{-2mm}
{\small Source: Smets and Wouters (2007) ``Shocks and Frictions in US Business Cycles"}\\[-0.5ex]
{\scriptsize\url{https://www.aeaweb.org/articles?id=10.1257/aer.97.3.586}}
\end{frame}





\begin{frame}{Next lecture: policy in the New Keynesian model}
\be
\item Monetary policy
\bi
\medskip
\item already part of model, can use as is
\ei
\bigskip
\item Fiscal policy
\bi
\medskip
\item not in model yet, will have to extend it
\ei
\ee
\bigskip
Key observation: \tr{sticky prices break first welfare theorem}
\bi
\medskip
\item sticky prices = ``friction"
\medskip
\item $\Rightarrow$ rationalizes some sort of \tr{policy intervention}
\ei
\end{frame}



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