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\title{\vspace{1.6cm} \\
	{\LARGE Lecture 9}\\[1ex]
{\LARGE The Financial Crisis, Asset Bubbles}\\
	\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}{} 
\begin{frame}[noframenumbering]
\titlepage
\end{frame}
}

\begin{frame}{Plan}
\be
%\bigskip
%\item 2023 Run on Silicon Valley Bank 
\bigskip
\item The 2008 financial crisis: some facts 
\bigskip
\item Asset bubbles 
\bigskip
\item Financial frictions and amplification 
\ee
\end{frame}

\begin{frame}{A Nobel Prize for Work on Financial Crises}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.42\textwidth]{figures/bernanke-diamond-dybvig} 
\end{figure}
\vspace{-2mm}
Source: {\scriptsize \url{https://www.nobelprize.org/prizes/economic-sciences/2022/summary/}}
\end{frame}



\begin{frame}{Readings}
\vspace{-4mm}
\bi
\item Two supplements with derivations on moodle:
\bi
\smallskip
\item asset bubbles
\smallskip
\item financial frictions and amplification
\ei
\medskip
\item EC1B1 lecture notes 3 ``Great Depression \& Lender of Last Resort"
\medskip
\item Jones, chapter 10.4
\medskip
\item Additional readings for the interested (not examinable)
\bi
{\footnotesize
\smallskip
\item Brunnermeier (2008) ``Bubbles"\\[-1ex] {\tiny\url{https://link.springer.com/referenceworkentry/10.1057/978-1-349-95121-5_44-2}}
\smallskip
\item Bernanke and Gertler (1989), ``Agency Costs, Net Worth, and Business Fluctuations"
\smallskip
\item Kiyotaki and Moore (1997), "Credit Cycles"
\smallskip
\item Mian and Sufi (2011) ``House Prices, Home Equity-Based Borrowing, and the U.S. Household Leverage Crisis" (2011)
\smallskip
\item Mian and Sufi (2014) ``What Explains the 2007–2009 Drop in Employment?"
%\item Noah Smith on SVB {\tiny \url{https://noahpinion.substack.com/p/why-was-there-a-run-on-silicon-valley}}
%\item Adam Tooze on SVB {\tiny \url{https://adamtooze.substack.com/p/chartbook-200-something-broke-the}}
%\item Jiang et al (2023) ``Monetary Tightening and U.S. Bank Fragility in 2023"\\[-1ex] {\tiny \url{https://www.nber.org/papers/w31048}}
}
\ei
\ei
\end{frame}

%
%
%\begin{frame}
%\renewcommand{\baselinestretch}{1}
%\vspace{3cm}
%\begin{center}
%{\huge 2023 Run on Silicon Valley Bank}
%\end{center}
%\end{frame}
%
%
%
%\begin{frame}{Bank runs as multiple equilibria (from EC1B1)}
%\vspace{-4mm}
%\begin{figure}[tph]
%	\centering
%\includegraphics[width=\textwidth]{figures/EC1B1_bank_runs}
%\includegraphics[width=\textwidth]{figures/EC1B1_bank_runs2}
%\end{figure}
%\end{frame}
%
%
%\begin{frame}{Deposit insurance}
%\bi
%\item Solution? Deposit insurance
%\bigskip
%\item In theory on previous slide:
%\bi
%\medskip
%\item eliminates bad equilibrium
%\medskip
%\item never even has to be paid out
%\ei
%\bigskip
%\item In U.S., Federal Deposit Insurance Corporation (FDIC) insures all deposits up to \$250,000
%\ei
%\end{frame}
%
%\begin{frame}{This works not only in theory but also in practice}
%\vspace{-4mm}
%\begin{figure}[tph]
%	\centering
%\includegraphics[width=.75\textwidth]{figures/wolfers-FDIC} 
%\end{figure}
%\end{frame}
%
%
%\begin{frame}{{\large But most SVB accounts were $>\$ 250,000$ so not FDIC insured}}
%\vspace{-4mm}
%\begin{figure}[tph]
%	\centering
%\includegraphics[width=.75\textwidth]{figures/wolfers-SVB} 
%\end{figure}
%\end{frame}
%
%
%\begin{frame}{{\large But most SVB accounts were $>\$ 250,000$ so not FDIC insured (zoom)}}
%\vspace{-4mm}
%\begin{figure}[tph]
%	\centering
%\includegraphics[width=.8\textwidth]{figures/wolfers-SVB-zoom} 
%\end{figure}
%\end{frame}
%
%
%
%\begin{frame}{A ``bank run via whatsapp"}
%\vspace{-4mm}
%One account of the run {\notsotiny \url{https://twitter.com/torrenegra/status/1634573234187407369?s=20}}
%\begin{figure}[tph]
%	\centering
%\includegraphics[width=.4\textwidth]{figures/whatsapp-run} 
%\end{figure}
%\end{frame}
%
%
%
%\begin{frame}{SVB's other big problem: unhedged interest rate risk}
%\vspace{-2mm}
%Recall typical bank balance sheet from EC1B1
%\begin{figure}[tph]
%	\centering
%\includegraphics[width=\textwidth]{figures/bank_leverage} 
%\end{figure}
%But SVB also held large chunk of other assets: long-term fixed income securities
%\bi
%\smallskip
%\item e.g. 10-year government bonds
%\smallskip
%\item rising interest rate $\Rightarrow$ prices of such assets fall
%\smallskip
%\item SVB did not hedge this interest rate risk appropriately
%\ei
%\end{frame}
%
%
%\begin{frame}{Like all banks, SVB was also very leveraged}
%\vspace{-4mm}
%\begin{figure}[tph]
%	\centering
%\includegraphics[width=\textwidth]{figures/bank_leverage2} 
%\end{figure}
%\end{frame}
%
%
%
%\begin{frame}{Summary: run on Silicon Valley Bank}
%Two key features made SVB particularly vulnerable
%\be
%\bigskip
%\item large runnable deposit base
%\bigskip
%\item big interest rate risk
%\ee
%\bigskip
%\tr{SVB was special ... but definitely not alone} $\Rightarrow$ will have to wait and see
%\end{frame}
%



\begin{frame}
\renewcommand{\baselinestretch}{1}
\vspace{3cm}
\begin{center}
{\huge 2008 Financial Crisis: Some Facts}
\end{center}
\end{frame}



\begin{frame}{2008 financial crisis: some facts}
\bi
\item Will show you data for U.S. 
\bi
\medskip
\item purely because many nice graphs are available
\ei
\bigskip
\item UK and many other advanced economies look similar
\bi
\medskip
\item though some not as extreme
\ei
\ei
\end{frame}


\begin{frame}{Housing price boom and bust in lead up to crisis}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.9\textwidth]{figures/jones_house_prices}
\end{figure}
\end{frame}


\begin{frame}{A very large drop in GDP ...}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.9\textwidth]{figures/jones_GDP}
\end{figure}
\end{frame}

\begin{frame}{... and employment}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=\textwidth]{figures/jones_employment}
\end{figure}
\end{frame}

\begin{frame}
\renewcommand{\baselinestretch}{1}
\vspace{3cm}
\begin{center}
{\huge Asset Bubbles}
\end{center}
\end{frame}


\begin{frame}{What is a bubble?}
\bi
\item Price of an asset does not reflect its ``fundamental value"
\bigskip
\item Idea: speculators buy the asset only because they expect its price to rise in the future
\bigskip
\item Self-fulfiling expectations: ``prices rise because they are expected to rise"
\bigskip
\item Now: a simple model of a bubble
\ei
\end{frame}

\begin{frame}{A simple asset pricing model}
\bi
\item Asset pays dividends
$$\{y_t\}_{t=0}^\infty = y_0,y_1,...$$
\item Individuals discount future at $\beta=1/(1+r)$ satisfying $0<\beta<1$
\bigskip
\item How will this asset be priced?
\bigskip
\item Main example: asset = house
\bi
\medskip
\item dividend $y_t$ =  per-period benefit received from owning the house
\medskip
\item either rent (investment) or benefit from living in house (owner-occupied)
\medskip
\item will sometimes say ``rent" instead of ``dividend"
\ei
\ei
\end{frame}

\begin{frame}{A difference equation for pricing the asset}
\vspace{-4mm}
\bi
\item \tr{\textbf{Claim:}} the asset price $p_t$ must satisfy the difference equation
\begin{equation}\label{eq:arbitrage}\tag{$\ast$}
\tr{p_t = y_t + \beta p_{t+1}}
\end{equation}
In words: price today = dividend + discounted price tomorrow
\medskip
\item \tr{\textbf{Intuition:}} arbitrage -- see supplement
\medskip
\item \tr{\textbf{Claim:}} A solution to the difference equation \eqref{eq:arbitrage} is
\begin{equation}\label{eq:nobubble}\tag{$\ast \ast$}
p_t = \sum_{j=0}^\infty \beta^j y_{t+j} = y_t + \beta y_{t+1} + \beta^2 y_{t+2} + \beta^3 y_{t+3}...
\end{equation}
In words: Price = present discounted value (PDV) of future dividends
\medskip
\item \tr{\textbf{Example:}} constant dividend $y_t = \bar{y}$ for all $t$ (using $\sum_{j=0}^\infty \beta^j = 1/(1-\beta)$)
$$p_t = \frac{\bar{y}}{1-\beta} $$
\vspace{-6mm}
\item Is this a bubble? No
\smallskip
\item PDV of future dividends = correct notion of fundamental value
\ei
\end{frame}



\begin{frame}{A difference equation for pricing the asset}
\vspace{-4mm}
\bi
\item Recall: the asset price $p_t$ must satisfy the difference equation
\begin{equation}\label{eq:arbitrage}\tag{$\ast$}
\tr{p_t = y_t + \beta p_{t+1}}
\end{equation}
with one solution given by
\begin{equation}\label{eq:nobubble}\tag{$\ast \ast$}
p_t = \sum_{j=0}^\infty \beta^j y_{t+j} = y_t + \beta y_{t+1} + \beta^2 y_{t+2} + \beta^3 y_{t+3}...
\end{equation}
\item How can bubbles arise?
\medskip
\item \tr{\textbf{Key:}} asset price $p_t$ in \eqref{eq:nobubble} is not \tr{unique} solution to \eqref{eq:arbitrage}
\bi
\item equation \eqref{eq:arbitrage} has many more solutions that all correspond to bubbles
\ei
\ei
\end{frame}


\begin{frame}{A useful case: zero dividends $y_t = 0$}
\vspace{-4mm}
\bi
\item Asset pricing equation is
$$p_t = \beta p_{t+1} $$
\item Obvious solution 
$$p_t = 0 = \textrm{no-bubble solution} $$
\item But there is another solution
$$p_t = c \left(\frac{1}{\beta} \right)^t = c \beta^{-t} \quad \mbox{for a constant} \ c $$
\item Verify:
$$p_t = c \beta^{-t} = \beta \underbrace{[c \beta^{-(t+1)}]}_{p_{t+1}} = \beta p_{t+1} $$
\item In fact this works for \tr{any} constant $c$ so there are infinitely many such solutions
\medskip
\item All these solutions are bubble solutions: $p_t \neq$ fundamental value = 0 
\ei
\end{frame}

\begin{frame}{A pure bubble when the asset pays no dividend $y_t = 0$}
\vspace{-3mm}
When $c>0$ asset price $p_t = c \beta^{-t}$ grows exponentially (recall $\beta<1$)
\vspace{-1mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.6\textwidth]{figures/lec9_2} 
\end{figure}
\vspace{-4mm}
\bi
\item Buy worthless asset because expect to sell it at higher price tomorrow
\smallskip
\item ``Prices rise because they are expected to rise"
\ei
\end{frame}


\begin{frame}{Putting everything together: positive dividends $y_t$}
\bi
\item Recall: the asset price $p_t$ must satisfy the difference equation
\begin{equation}\label{eq:arbitrage}\tag{$\ast$}
p_t = y_t + \beta p_{t+1}
\end{equation}
\item \tr{\textbf{Claim:}} general solution to \eqref{eq:arbitrage} is
$$p_t = \underbrace{\sum_{j=0}^\infty \beta^j y_{t+j}}_{\mbox{\footnotesize fundamental value}} + \underbrace{c \beta^{-t}}_{\mbox{\footnotesize bubble component}} \quad \mbox{for \tr{any} constant} \ c$$
\item Next slide: graph with constant dividend and hence constant fundamental value
\ei
\end{frame}





\begin{frame}{Asset price = fundamental value + bubble component}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.8\textwidth]{figures/lec9_1} 
\end{figure}
\end{frame}



\begin{frame}{Asset bubbles: summary}
\bi
\item It's not necessarily true that
$$\mbox{asset price} = \mbox{fundamental value} = \mbox{PDV of dividends}$$
\item Instead there can be self-fulfilling bubbles in which
$$\mbox{asset price} = \mbox{fundamental value} + \mbox{bubble component}$$
\item These bubbles are self-fulfiling in sense that prices rise because they are expected to rise
\bigskip
\item For more on bubbles, see survey by Brunnermeier (2008) ``Bubbles" in the The New Palgrave Dictionary of Economics \\
{\scriptsize\url{https://link.springer.com/referenceworkentry/10.1057/978-1-349-95121-5_44-2}}
\ei
\end{frame}


\begin{frame}
\renewcommand{\baselinestretch}{1}
\vspace{3cm}
\begin{center}
{\huge Financial Frictions and Amplification }
\end{center}
\end{frame}



\begin{frame}{The idea in a nutshell: ``financial doom loop"}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.7\textwidth]{figures/loop} 
\end{figure}
\bi
\item Sometimes people call this ``financial doom loop" or ``diabolic loop"
\medskip
\item An ``LSE idea": Key work by Nobu Kiyotaki and John Moore written while both were at LSE (John Moore still is)
\ei
\end{frame}



\begin{frame}{Watch the interviews here, particularly John Moore's}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.75\textwidth]{figures/BBVA} 
\end{figure}
\vspace{-4mm}
{\tiny \url{https://www.frontiersofknowledgeawards-fbbva.es/noticias/the-frontiers-of-knowledge-award-goes-to-bernanke-gertler-kiyotaki-and-moore-for-establishing-the-nature-of-the-linkage-between-the-financial-sector-and-the-real-economy-and-how-it-operates/}}

\end{frame}



\begin{frame}{Plan for this part of lecture notes}
\be
\item Preliminaries: Borrowing and Saving in a Small Open Economy
\bigskip
\item Preliminaries: Consumption Based Asset Pricing
\bigskip
\item Borrowing and Saving with a ``Collateral Asset": Financial Amplification
\ee
\bigskip
Here:
\bi
\medskip
\item due to time constraints only provide overview
\medskip
\item see supplement for more detailed version (examinable)
\ei
\end{frame}



\begin{frame}{Borrowing and Saving in a Small Open Economy}
\vspace{-3mm}
\bi
\item ``Small open economy": household can borrow and lend at a fixed world interest rate $r^*$
\medskip
\item Consumers solve
\vspace{-3mm}
\begin{align*}\max_{c_1,c_2,d_1} \ &u(c_1) + \beta u(c_2) \quad \mbox{s.t.} \\
&c_1 = y_1 + d_1 \\
&c_2 + d_1(1+r^*) = y_2\\
&\tr{d_1 \leq \kappa y_1}, \quad \quad \kappa \geq 0
\end{align*}
\item Note: $d_1$ is debt, i.e. how much household borrows in period 1
\medskip
\item Note presence of \tr{borrowing constraint $d_1 \leq \kappa y_1$}
\bi
\smallskip
\item borrow up to fraction (or multiple) $\kappa$ of first-period income $y_1$
\smallskip
\item $\kappa$ parameterizes quality of credit markets
\smallskip
\item $\kappa = \infty$: can borrow as much as you'd like (no constraint)
\smallskip
\item $\kappa = 0$: cannot borrow at all
\ei
\ei
\end{frame}



\begin{frame}{Unconstrained solution $\kappa = \infty$}
\vspace{-3mm}
\bi
\item Can write present-value budget constraint
$$c_1 + \frac{c_2}{1+r^*} = y_1 + \frac{y_2}{1+r^*} \equiv y^{PDV}$$
\item Optimality condition = standard Euler equation
$$u'(c_1) = \beta(1+r^*)u'(c_2)$$
\item Assumption: $\beta(1+r^*) = 1$
\bigskip
\item Then unconstrained solution is
$$c_1^u = c_2^u = \frac{1+r^*}{2+r^*}y^{PDV}, \qquad d_1^u = c_1^u - y_1 = \frac{y_2 - y_1}{2+r^*}$$
where $u$-subscript stands for ``unconstrained"
\ei
\end{frame}




\begin{frame}{Solution with borrowing constraint  $d_1 \leq \kappa y_1$ with $\kappa < \infty$}
\vspace{-2mm}
\tr{\textbf{Case 1: $d_1^u \leq \kappa y_1$ (loose constraint)}}
\bi
\medskip
\item \tr{can} obtain unconstrained allocation $(c_1^u,c_2^u,d_1^u)$
\medskip
\item = optimal choice and constraint will never bind
\ei
\bigskip
\tr{\textbf{Case 2: $d_1^u > \kappa y_1$ (binding constraint)}}
\bi
\medskip
\item \tr{cannot} obtain unconstrained allocation $(c_1^u,c_2^u,d_1^u)$
\medskip
\item household will borrow as much as it can $d_1 = \kappa y_1$ and consumption is
$$c_1 = (1+\kappa)y_1 , \quad c_2 = y_2 - \kappa y_1(1+r^*)$$
\item $c_1 < c_1^u = c_2^u < c_2$, i.e. can no longer smooth consumption perfectly
\medskip
\item borrowing constraint makes them strictly worse off
\ei
\end{frame}


\begin{frame}{A credit crunch $\kappa \downarrow$}
\bi
\item Recall
$$c_1 = (1+\kappa)y_1 , \quad c_2 = y_2 - \kappa y_1(1+r^*)$$
\item Therefore $\kappa \downarrow \Rightarrow c_1 \downarrow $ and $c_2 \uparrow$
\bigskip
\item Even worse consumption smoothing $\Rightarrow$ welfare falls more
\ei
\end{frame}



\begin{frame}{Consumption-based asset pricing}
\vspace{-3mm}
\bi
\item Now: no borrowing and lending but households can \tr{invest in an asset $a_t$}
\bi
\smallskip
\item buy asset at \tr{price $p_1$} in period $1$
\smallskip
\item asset pays a dividend $D$ in period $2$
\smallskip 
\item asset is in \tr{fixed supply} $a_0^s = a_1^s = 1$, i.e. there is one unit of the asset
\ei
\medskip
\item Households solve
\begin{align*}\max_{c_1,c_2,a_1} \ &u(c_1) + \beta u(c_2) \quad \mbox{s.t.} \\
&c_1 + p_1 a_1 = y_1 + p_1 a_0 \\
&c_2 = y_2 + D a_1
\end{align*}
\item In equilibrium $a_0 = a_1 = 1$ and hence already know ($^e$ for ``equilibrium")
$$c_1^e = y_1, \quad c_2^e = y_2 + D $$
\item Only question: \textbf{\tr{what is the equilibrium asset price $p_1$?}}
\bi
\smallskip
\item note similarity to finding equilibrium $r$ in last part of lecture 4
\ei
\ei
\end{frame}


\begin{frame}{Equilibrium asset price}
\vspace{-3mm}
\bi
\item Optimality condition
$$p_1 u'(c_1) = \beta D u'(c_2) $$
\item But we already know that in equilibrium
$$c_1^e = y_1, \qquad c_2^e = y_2 + D$$
\item Therefore equilibrium asset price is
$$p_1 = \frac{\beta u'(c_2^e)}{u'(c_1^e)} D = \frac{\beta u'(y_2 + D)}{u'(y_1)} D $$
\item Example: log utility $u(c) = \log c$
$$p_1 = \frac{\beta c_1^e}{c_2^e} D $$
\item Note: rather than asking ``given prices, what is consumption?" we asked ``given consumption, what is the price?"
\ei
\end{frame}

\begin{frame}{Equilibrium asset price $p_1$ is depressed when $c_1$ is low}
\vspace{-3mm}
\bi
\item Equilibrium asset price
$$p_1 = \frac{\beta u'(c_2^e)}{u'(c_1^e)} D = \frac{\beta u'(y_2 + D)}{u'(y_1)} D $$
\item Example: log utility $u(c) = \log c$
$$p_1 = \frac{\beta c_1^e}{c_2^e} D = \frac{\beta y_1}{y_2 + D} D $$
\item Interesting feature: $p_1$ is low when $c_1^e = y_1$ is low and marginal utility $u'(c_1^e)$ is high
\bigskip
\item Intuition: \tr{don't want to buy asset if you're starving $\Rightarrow$ low asset price}
\bigskip
\item This will be key feature of model we want to get to
\ei
\end{frame}


\begin{frame}{{\large Borrowing and saving with collateral asset: financial amplification}}
\vspace{-3mm}
\bi
\item Now combine elements from two preliminaries
\bi
\smallskip
\item borrowing and saving in small open economy
\smallskip
\item consumption-based asset pricing
\ei
\bigskip
\item Key new ingredient = \tr{collateral constraint}: borrowing constraint in which amount of debt $d_1$ is constrained by value of its assets
\bigskip
\item Here: assume
\tr{$$d_1 \leq \kappa p_1 a_0$$}
\item Idea: can borrow against value of existing assets (in case of housing: ``houses as ATMs", i.e. HELOC = home equity line of credit)
\bigskip
\item Alternative formulation: $d_1 \leq \kappa p_1 a_1$ -- see supplement
\ei
\end{frame}




\begin{frame}{{\large Borrowing and saving with collateral asset: financial amplification}}
\vspace{-3mm}
\bi
\item Households solve:
\begin{align*}\max_{c_1,c_2,a_1,d_1} \ &u(c_1) + \beta u(c_2) \quad \mbox{s.t.} \\
&c_1 + p_1 a_1 = y_1 + p_1 a_0 + d_1 \\
&c_2 + d_1(1+r^*) = y_2 + D a_1 \\
&\tr{d_1 \leq \kappa p_1 a_0} \end{align*}
\item  Asset is still in fixed supply: $a_0^s = a_1^s = 1$
\bigskip
\item Unconstrained solution: see supplement
\bigskip
\item As before, two cases: loose constraint and binding constraint
\bigskip
\item Here: focus on binding constraint
\ei
\end{frame}


\begin{frame}{Equilibrium with binding constraint}
\vspace{-3mm}
\bi
\item Borrow as much as possible, imperfect consumption smoothing
$$d_1 = \kappa p_1 a_0 = \kappa p_1, \quad c_1 = y_1 + \kappa p_1, \quad c_2 = y_2 + D - \kappa p_1(1+r^*) $$
\item Equilibrium asset price determined by
$$p_1 = \frac{\beta u'(c_2)}{u'(c_1)}D =  \frac{\beta u'(y_2 + D - \kappa p_1(1+r^*))}{u'(y_1 + \kappa p_1)}D$$
\item Key feature again: equilibrium asset price $p_1$ is depressed when $c_1$ is low
\bigskip
\item To solve this neatly change utility slightly: log-linear utility
$$u_1(c_1) + \beta u_2(c_2), \quad u_1(c_1) = \log c_1, \quad u_2(c_2) = c_2 $$
\item Also assume $\kappa \beta D<1$
\ei
\end{frame}


\begin{frame}{Equilibrium with binding constraint and log-linear utility}
\vspace{-3mm}
\bi
\item With log-linear utility equilibrium price satisfies
\begin{align*}
p_1 &= \beta D c_1 \\
c_1 &= y_1 + \kappa p_1
\end{align*}
\item Solving for $p_1$ and $c_1$
\tr{$$p_1 = \frac{\beta D y_1}{1-\beta D \kappa}, \qquad c_1 = \frac{y_1}{1-\beta D \kappa}  $$}
\item Equilibrium features \tr{financial amplification}
\bigskip
\item Suppose $y_1 \downarrow$. Unsurprisingly $c_1 \downarrow$
\bigskip
\item But key: $c_1$ may fall by a lot more than $y_1$!
$$\frac{\partial c_1}{\partial y_1} = \frac{1}{1-\beta D \kappa} \tr{>1} \quad \Rightarrow \quad \tr{\mbox{multiplier effect}} $$
\ei
\end{frame}


\begin{frame}{Intuition: ``financial doom loop"}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.45\textwidth]{figures/loop} 
\end{figure}
\bi
\item Key equations
\begin{align}
p_1 &= \beta D c_1 \\
c_1 &= y_1 + \kappa p_1
\end{align}
\item Mechanism: $y_1 \downarrow \Rightarrow$ consumption $c_1 \downarrow$ from (2) $\Rightarrow$ asset demand $\downarrow$ (``don't want to buy asset when starving") $\Rightarrow$ asset price $p_1 \downarrow$ from (1)  $\Rightarrow$ tighter collateral constraint $\Rightarrow$ $c_1 \downarrow$ from (2) $\Rightarrow p_1 \downarrow$ from (1) and so on...   
\ei
\end{frame}



\begin{frame}{Importance of house prices during the financial crisis}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.9\textwidth]{figures/mian-sufi.pdf} 
\end{figure}
\vspace{-2mm}
{\small Source: Mian and Sufi (2014) ``What Explains the 2007-2009 Drop in Employment?"}
\end{frame}



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