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\title{\vspace{1.6cm} \\
	{\LARGE Lecture 4}\\[1ex]
{\LARGE Consumption, Saving, Interest Rates}\\
	\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}{Today: your first \tr{dynamic} modern macro model}
\vspace{-3mm}
\bi
\item So far: static models = models without time
\medskip
\item But lots of questions in macro are inherently \tr{dynamic = they involve time}
\bi
\smallskip
\item Why do economies \tr{grow}? Why $\approx$ constant long-run \tr{growth rates}?
\smallskip
\item Why \tr{booms, busts}? Should policy \tr{stabilize} business \tr{cycles}? How?
\smallskip
\item In typical recession, why does $I$ \tr{decline} more than $C$?
\smallskip
\item even true for questions we've already covered, e.g. Germany without Russian gas $\Rightarrow$  importance of time horizon for (le Chatelier)
\smallskip
\item ... but so far we discussed such issues ``outside of the model"
\ei
\medskip
\item \textbf{\tr{Today and rest of course:}} there is time $t=0,1,2,...$ (typically discrete)
\medskip
\item Have already seen a dynamic model = Solow model, but no microfoundations (so $\neq$ modern macro)
\medskip
\item Also recall Keynesian IS-MP-PC model from EC1B1: $t$-subscripts but not really a proper dynamic model. And definitely not microfounded.
\ei
\end{frame}



\begin{frame}{IS-MP-PC model from EC1B1: words tell dynamic story, but then why are we just shifting around these curves?}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.92\textwidth]{figures/EC1B1_IS_shock.pdf} 
\end{figure}
\end{frame}


\begin{frame}{Plan}
\be
\item The Keynesian view of consumption
\bigskip
\item A two-period model of consumption
\bigskip
\item The permanent income hypothesis
\bigskip
\item Consumption, saving and interest rates in general equilibrium
\ee
\end{frame}



\begin{frame}{Background Readings}
\bi
\bigskip
\item Follow closely Kurlat Chapter 6, except part 4 on previous slide
\bigskip
\item EC1A1 lecture notes on consumption and saving (intertemporal choice)
\bigskip
\item EC1B1 notes on Keynesian consumption function and Keynesian cross
\ei
\end{frame}



\begin{frame}
\renewcommand{\baselinestretch}{1}
\vspace{3cm}
\begin{center}
{\huge The Keynesian view of consumption}
\end{center}
\end{frame}


\begin{frame}{Keynes on consumption (also see EC1B1 notes)}
\vspace{-3mm}
\bi
\item Keynes in his 1936 ``General Theory of Interest, Employment \& Money":
 {\small ``The fundamental psychological law, upon which we are entitled to depend with great confidence both a priori from our knowledge of human nature and from the detailed facts of experience, is that \tr{men are disposed}, as a rule and on the average, \tr{to increase their consumption as their income increases but not by as much as the increase in the income}."}
\bigskip
\pause
\item In equations, consumption $C$ depends on income $Y$ as
$$C = c(Y) \quad \mbox{with} \quad c'(Y)> 0 \quad \mbox{and} \quad c'(Y) < 1 $$
\item $c(Y)$ = consumption function. Question for you: name for $c'(Y)$?
\medskip
\item Depending on how interpret language, he may also mean
$$\frac{c'(Y)Y}{c(Y)} = \frac{\partial \log c(Y)}{\partial \log Y} < 1 $$
i.e. elasticity of $C$ with respect to $Y$ $<1$
\ei
\end{frame}



\begin{frame}{Importance of MPC for multipliers (Keynesian cross)}
\vspace{-3mm}
From EC1B1 notes: MPC = $\gamma$, consumption function $C = \alpha + \gamma(Y - T)$ \\
\vspace{-2mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.95\textwidth]{figures/EC1B1_keynesian_cross.pdf} 
\end{figure}
\end{frame}



\begin{frame}{In micro data both $c'(Y)$ and $\frac{\partial \log c(Y)}{\partial \log Y}$ are considerably $<1$}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=\textwidth]{figures/kurlat_fig6_1_1.pdf} 
\end{figure}
\end{frame}


\begin{frame}{Implications for aggregate consumption and income}
\bi
\item Recall: in micro data both $c'(Y)$ and $\frac{\partial \log c(Y)}{\partial \log Y}$ are considerably $<1$
\bigskip
\item Around 1950s: seen as conclusive evidence in favor of Keynesian consumption function
\bigskip
\item But then what happens as countries grow richer over time and $Y$ grows?
\bigskip
\item If $\frac{\partial \log c(Y)}{\partial \log Y}< 1$ then we should have
$$\frac{c(Y)}{Y} \quad \mbox{declining with} \ Y $$
so people consume proportionately less and less as country grows richer
\bigskip
\item Do we see this? No.
\ei
\end{frame}


\begin{frame}{In macro data $\frac{\partial \log c(Y)}{\partial \log Y}$ is much larger, $\frac{C}{Y}$  stable over time}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=\textwidth]{figures/kurlat_fig6_1_2.pdf} 
\end{figure}
\end{frame}


\begin{frame}{Summary}
\bi
\item In micro data both $c'(Y)$ and $\frac{\partial \log c(Y)}{\partial \log Y}$ are considerably $<1$
\bigskip
\item but in macro data, $\frac{\partial \log c(Y)}{\partial \log Y}$ is much larger (perhaps even $\approx 1$), $\frac{C}{Y}$  stable over time
\bigskip
\item Seems puzzling, what's going on?
\bigskip
\item Next: use economic theory to resolve puzzle
\ei
\end{frame}



\begin{frame}
\renewcommand{\baselinestretch}{1}
\vspace{3cm}
\begin{center}
{\huge A two-period model of consumption}
\end{center}
\end{frame}





\begin{frame}{A two-period model of consumption}
\vspace{-3mm}
\bi
\item For now: study problem of individual in isolation taking as given prices (partial equilibrium)
\bigskip
\item In a bit: general equilibrium
\bigskip
\item Two time periods $t=1$ and $t=2$
\medskip
\item Consumption $c_1$ and $c_2$, income $y_1$ and $y_2$
\medskip
\item Utility function
$$u(c_1) + \beta u(c_2) $$
with $u$ strictly increasing, concave, discount factor $0<\beta<1$
\bigskip
\item Vocabulary that sometimes comes up: discount \tr{factor} vs discount \tr{rate}
\bi
\smallskip
\item discount \tr{factor}: $\beta$, typically a bit below one
\smallskip
\item discount \tr{rate}: $\rho$, typically a bit above zero
\smallskip
\item link: $\rho = 1/\beta - 1 \Leftrightarrow \beta = \frac{1}{1+\rho}$, e.g. $\beta=0.95,\rho = 1/0.95-1 = 5.26\%$
\ei
\ei
\end{frame}





\begin{frame}{A two-period model of consumption}
Household solves
\begin{align}
\notag \max_{c_1,c_2,a} \ u(c_1) & + \beta u(c_2) \quad \mbox{s.t.} \\
\tag{1}c_1 + a&= y_1\\
\tag{2}c_2&= y_2 + (1+r)a
\end{align}
Notation:
\bi
\medskip
\item $c_1,c_2$: consumption at $t=1$ and $t=2$
\medskip
\item $y_1,y_2$: income at $t=1$ and $t=2$
\medskip
\item $r$: interest rate (for now exogenously given)
\medskip
\item $a$: saving
\medskip
\item Note: $a$ can be negative, $a<0$ means household is borrowing
\medskip
\item From (1) $a<0 \Rightarrow c_1>y_1$, i.e. consume more than income by borrowing
\ei
\end{frame}

\begin{frame}{Formulation in terms of present-value budget constraint}
\bi
\item Implicit assumption: can borrow and save as much you want at rate $r$
\medskip
\item then can combine (1) and (2) into present-value budget constraint
$$
c_1 + a= y_1 \quad \mbox{and} \quad a= \frac{c_2}{1+r} - \frac{y_2}{1+r} \quad \Rightarrow \quad \underbrace{c_1 + \frac{c_2}{1+r}}_{\textrm{PV of consumption}} = \underbrace{y_1 + \frac{y_2}{1+r}}_{\textrm{PV of income}}
$$
where PV stands for present value
\bigskip
\item Hence households maximize utility s.t. present-value budget constraint
\ei
\begin{align*}
\max_{c_1,c_2} \ u(c_1) & + \beta u(c_2) \quad \mbox{s.t.} \\
c_1 + \frac{c_2}{1+r}&= y_1 + \frac{y_2}{1+r}
\end{align*}
\end{frame}


\begin{frame}{Euler equation}
\bi
\item Optimality condition
\begin{equation}\label{eq:EE}\tag{$\ast$}
\tr{u'(c_1) = \beta(1+r) u'(c_2) }
\end{equation}
\item Intuition: consume \$1 now vs save and consume later
\bi
\medskip
\item consume \$1 now: utility increases by $u'(c_1)$
\medskip
\item save \$1: have $\$ (1+r)$ at $t=2$, utility by $\beta u'(c_2)$ for each \$
\ei
\bigskip
\item \eqref{eq:EE} is called ``Euler equation" after Leonard Euler, plays an important role in macroeconomics
\bigskip
\item Note: ``Euler equation" sounds fancy but simply means ``intertemporal optimality condition"
\ei
\end{frame}


\begin{frame}{Graphical representation}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=\textwidth]{figures/kurlat_fig6_2_1.pdf} 
\end{figure}
\end{frame}

\begin{frame}{A saver (left panel) and a borrower (right panel)}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=\textwidth]{figures/kurlat_fig6_2_2.pdf} 
\end{figure}
Left panel: $y_2 = 0$ and $y_1> 0$ $\Rightarrow$ save. Right panel: $y_1 << y_2$ $\Rightarrow$ borrow.
\end{frame}


\begin{frame}{What happens when $r$ changes?}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=\textwidth]{figures/kurlat_fig6_2_3.pdf} 
\end{figure}
\end{frame}

\begin{frame}{Effect of change in $r$ for saver (left) and borrower (right)}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=\textwidth]{figures/kurlat_fig6_2_4.pdf} 
\end{figure}
\end{frame}



\begin{frame}
\renewcommand{\baselinestretch}{1}
\vspace{3cm}
\begin{center}
{\huge The permanent income hypothesis}
\end{center}
\end{frame}

\begin{frame}{An important observation about our model}
\vspace{-4mm}
\bi
\item Can write two-period model as
\begin{align*}
\max_{c_1,c_2} \ u(c_1) & + \beta u(c_2) \quad \mbox{s.t.}  \quad c_1 + \frac{c_2}{1+r}= \tr{W}
\end{align*}
$$\mbox{where} \quad \tr{W = y_1 + \frac{y_2}{1+r}} = \mbox{PV of income}$$
sometimes also called \tr{``lifetime income"} or \tr{``permanent income"}
\medskip
\item Can immediately see: $c_1$ will depend on $y_1,y_2$ only through $W$, i.e.
$$c_1 = c(W), \quad W = y_1 + \frac{y_2}{1+r} $$
\item It's not current income that matters, but PV of current + future income
\medskip
\item This way of thinking: \tr{``permanent income hypothesis"} (Friedman, 1957)
\medskip
\item Contrast with Keynes quote, captured w consumption function $c_1 = c(y_1)$
\medskip
\item Can already see: our model is really very different from Keynes view
\medskip
\item Next: flesh this out with parametric example
\ei
\end{frame}




\begin{frame}{A parametric example}
\vspace{-7mm}
\begin{equation}\label{eq:CRRA}\tag{$\ast \ast$}
u(c) = \frac{c^{1-\frac{1}{\sigma}}-1}{1-\frac{1}{\sigma}}
\end{equation}
\vspace{-3mm}
\bi
\item $\sigma$ is called ``intertemporal elasticity of substitution (IES)"
\medskip
\item Sometimes $1/\sigma$ is called ``coefficient of relative risk aversion (RRA)" and \eqref{eq:CRRA} is called ``CRRA utility"
\medskip
\item Can show: log utility = special case with $\sigma = 1$ (use l'Hopital's rule)
\vspace{-1mm}
$$u(c) = \lim_{\sigma \rightarrow 1}\frac{c^{1-\frac{1}{\sigma}}-1}{1-\frac{1}{\sigma}} = \log c$$
\item Note: weird $-1$ in numerator only there because we want to take this limit
\medskip
\item Important for those reading Kurlat: he, many others use $\sigma$ for coeff of RRA
$$u(c) = \frac{c^{1-\sigma}}{1-\sigma} \quad \mbox{or} \quad u(c) = \frac{c^{1-\sigma}-1}{1-\sigma} $$
\item Same but $\sigma$ in place of $\frac{1}{\sigma}$. Unfortunate notation, but don't get confused!
\ei
\end{frame}



\begin{frame}{Euler equation in parametric example}
\bi
\item With functional form \eqref{eq:CRRA} have
$$u'(c) = c^{-1/\sigma} $$
\item Euler equation
$$u'(c_1) = \beta(1+r) u'(c_2) \quad \Rightarrow \quad c_1^{-1/\sigma} = \beta(1+r) c_2^{-1/\sigma} $$
or
$$\tr{\frac{c_2}{c_1} = [\beta(1+r)]^\sigma} $$
\item IES $\sigma$ governs responsiveness of growth rate of consumption $c_2/c_1$ to changes in $r$ and $\beta$, e.g.
$$\frac{\partial \log (c_2/c_1)}{\partial \log (1+r)} = \sigma \quad \Rightarrow \mbox{hence the name IES} $$
\item A low IES $\sigma$ means households dislike intertemporal substitution (want to smooth $c$) $\Rightarrow$ low responsiveness of $c_2/c_1$ to changes in $r$ and $\beta$
\ei
\end{frame}


\begin{frame}{Analytic solution in parametric example}
\bi
\item Not hard to show (you should be able to!): solution to problem is
$$
c_1 = \frac{\left(\frac{1}{\beta (1+r)} \right)^\sigma (1+r)}{1+\left(\frac{1}{\beta (1+r)} \right)^\sigma (1+r)} W, \qquad c_2 =  \frac{1+r}{1+\left(\frac{1}{\beta (1+r)} \right)^\sigma (1+r)} W
$$
where 
$$W = y_1 + \frac{y_2}{1+r}$$
\medskip
\item Useful special case (see next slide why): $\beta(1+r)=1$
$$c_1 = c_2 = \frac{1+r}{2+r}W $$
\item If further $r=0$, then $c_1=c_2 = \frac{W}{2}$
\ei
\end{frame}



\begin{frame}{Reasonable parameterizations}
Reasonable parameterizations have two features
\be
\item $\beta(1+r)$ not too far from $1$, equivalently discount rate $\rho = 1/\beta-1 \approx r$
\bi
\medskip
\item e.g. $\rho =0.05,r=0.01$ but not $\rho= 0.5,r=0.01$ or $\rho = 0.05,r=0.5$
\medskip
\item in general equilibrium model of part 4, $1+r^* \approx 1/\beta$
\medskip
\item market discounts future at roughly the same rate as individuals
\ei
\bigskip
\item $\sigma$ not too large
\bi
\medskip
\item e.g. $\sigma=1/2$, probably $\leq 1$, definitely not $\sigma = 5$ or $10$
\medskip
\item empirical evidence: people do not massively substitute intertemporally, i.e. they do not massively increase spending when $r \downarrow$
\ei
\ee
\end{frame}



\begin{frame}{MPC out of transitory income shock}
\vspace{-4mm}
\bi
\item Question: suppose $y_1$ increases by \$1 but $y_2$ is unchanged. By how much does $c_1$ increase? What is MPC out of \tr{transitory} income shock?
\bigskip
\item Answer: the marginal propensity to consume (MPC) out of a transitory income shock is
$$\frac{\partial c_1}{\partial y_1} = \frac{\left(\frac{1}{\beta (1+r)} \right)^\sigma (1+r)}{1+\left(\frac{1}{\beta (1+r)} \right)^\sigma (1+r)} $$
\item \tr{Contrast with Keynesian cross:} $C = \alpha + \gamma (Y-T)$, $\gamma$ = exogenously given
\medskip
\item In current model, MPC is instead endogenous and depends on preferences $(\beta,\sigma)$ and prices $(r)$!
\medskip
\item This is precisely what we mean when we say ``Keynesian cross is not microfounded but modern macro models are"
\ei
\end{frame}




\begin{frame}{MPC out of transitory income shock}
\vspace{-4mm}
\bi
\item Question: suppose $y_1$ increases by \$1 but $y_2$ is unchanged. By how much does $c_1$ increase? What is MPC out of \tr{transitory} income shock?
\bigskip
\item Answer: the marginal propensity to consume (MPC) out of a transitory income shock is
$$\frac{\partial c_1}{\partial y_1} = \frac{\left(\frac{1}{\beta (1+r)} \right)^\sigma (1+r)}{1+\left(\frac{1}{\beta (1+r)} \right)^\sigma (1+r)} $$
\item Also note: \tr{this MPC is a number $<1$} and $<<1$ for reasonable parameters\\
\medskip
\item In fact, in this 2-period model, MPC $\approx 1/2$, e.g. with $\beta(1+r)=1$
$$\frac{\partial c_1}{\partial y_1} = \frac{1+r}{2+r}$$
and if $r=0$ then $c_1 = W/2$ and 
$$\frac{\partial c_1}{\partial y_1} = \frac{1}{2}$$
\ei
\end{frame}





\begin{frame}{MPC out of permanent income shock}
\vspace{-3mm}
\bi
\item Question: suppose both $y_1$ and $y_2$ increase by \$1. By how much does $c_1$ increase? What is MPC out of \tr{permanent} income shock?
\bigskip
\item Assumption: $y_1 = \bar{y_1} + \Delta,y_2 = \bar{y}_2 + \Delta$ and $\Delta$ increases
\bigskip
\item From $W = y_1 + \frac{y_2}{1+r}$: when both $y_1,y_2 \uparrow$ by \$1, $W \uparrow$ by $\approx \$2$
$$\frac{\partial W}{\partial \Delta} = 1 + \frac{1}{1+r} = \frac{2+r}{1+r} \approx 2$$
\item Similarly
$$\frac{\partial c_1}{\partial \Delta} = \frac{\left(\frac{1}{\beta (1+r)} \right)^\sigma (2+r)}{1+\left(\frac{1}{\beta (1+r)} \right)^\sigma (1+r)} $$
\item For reasonable parameters \tr{this is a number $\approx 1$}. Exact when $\beta(1+r)=1$
$$\frac{\partial c_1}{\partial \Delta} = 1$$
\ei
\end{frame}




\begin{frame}{Summary: main prediction of permanent income hypothesis}
\bi
\item \tr{Main prediction of permanent income hypothesis (PIH):}\\[2ex]
 MPC out of transitory income $<<$ MPC out of permanent income
\bigskip
\bigskip
\item Intuition: save large fraction of transitory income change so as to smooth consumption over time
\ei
\end{frame}








\begin{frame}{Extension to many time periods}
\vspace{-4mm}
\bi
\item Macro models used for research, policy etc: not just two time periods $t=1,2$. Instead: many time periods $t=1,2,3,...,T$
\bi
\item consumption $c_1,c_2,c_3,...,c_T$, income $y_1,y_2,y_3,...,y_T$
\item relatively straightforward extension -- see Kurlat chapter 6.3
\ei
\smallskip
\item \tr{Takeaway: more time periods make PIH prediction even starker}
\medskip
\item Key: PV budget constraint 
$$c_1 + \frac{c_2}{1+r} + \frac{c_3}{(1+r)^2} + ... + \frac{c_T}{(1+r)^{T-1}} = W$$
$$\mbox{where} \quad W = y_1 + \frac{y_2}{1+r} + \frac{y_3}{(1+r)^2} + ... + \frac{y_T}{(1+r)^{T-1}} $$
\item \tr{Makes PIH prediction even starker:} effect of transitory change in $y_1$ (say) has negligible effect on permanent income $W$, hence $\partial c_1/\partial y_1$ very small
\smallskip
\item Example: if $\beta(1+r)=1$ and $r=0$, then $c_1 = c_2 = ... = W/T$ and 
\vspace{-2mm}
$$\frac{\partial c_1}{\partial y_1} = \frac{1}{T} = \textrm{small number}$$
\ei
\end{frame}




\begin{frame}{Friedman's example: paydays staggered throughout week}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.5\textwidth]{figures/friedman.pdf}
\end{figure}
\vspace{-2mm}
Source: Friedman (1957) ``A Theory of the Consumption Function"\\
\vspace{-2mm}
{\tiny \url{https://www.nber.org/books-and-chapters/theory-consumption-function}}\\
This excerpt from conclusion {\tiny \url{https://www.nber.org/system/files/chapters/c4411/c4411.pdf}}
\end{frame}



\begin{frame}{The PIH predicts that multipliers are small}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.7\textwidth]{figures/EC1B1_keynesian_cross.pdf} 
\end{figure}
\vspace{-2mm}
PIH predicts that aggregate MPC out of current income is small, say $\gamma = 0.05$ \\
\medskip
$\Rightarrow$ Multipliers are small
\bi
\smallskip
\item government purchases multiplier $\approx 1$, say $=\frac{1}{1-0.05} = 1.053$
\smallskip
\item tax cut multiplier close $\approx 0$, say $=\frac{0.05}{1-0.05} = 0.053$
\ei
\end{frame}



\begin{frame}{Resolving our puzzle with the permanent income hypothesis}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.53\textwidth]{figures/kurlat_fig6_1_1.pdf} \includegraphics[width=.53\textwidth]{figures/kurlat_fig6_1_2.pdf} 
\end{figure}
\end{frame}


\begin{frame}{Resolving our puzzle with the permanent income hypothesis}
\tr{\textbf{Micro data:}}
\bi
\medskip
\item Transitory income shocks are important, some people have good year, others have bad year. Get mixed together with permanent income shocks.
\medskip
\item Coefficient in regression of $C$ on $Y$ reflects MPCs out of both permanent and transitory income shocks
\medskip
\item Estimate $c'(Y)$ and $\frac{\partial \log c(Y)}{\partial \log Y}$ considerably $<1$
\ei
\bigskip
\tr{\textbf{Macro data:}}
\bi
\medskip
\item Transitory income shocks mostly wash out in the aggregate
\medskip
\item Coefficient in regression of $C$ on $Y$ mostly reflects MPCs out of permanent income changes (e.g. country as a whole is richer on average)
\medskip
\item Estimate $c'(Y)$ and $\frac{\partial \log c(Y)}{\partial \log Y}$ much closer to $1$
\ei
\end{frame}



\begin{frame}{Empirical failure of the PIH}
\bi
\item Multi-period versions of the PIH generate MPCs out of transitory income shocks that are \tr{way too small} relative to data
\bigskip
\item Can show: model MPC $\approx r$ so something like 0.02 or 0.05 per year
\bigskip
\item In contrast, empirical MPCs are around 0.3 per year on average (i.e. people consume 30 cents for every \$1)
\bigskip
\item ... lots of heterogeneity in MPCs
\bigskip
\item ... and this heterogeneity depends on things like liquid wealth, i.e. more cash-strapped households have higher MPCs
\bigskip
\item See evidence on next slides
\bigskip
\item So perhaps Keynes wasn't as wrong as we thought?
\ei
\end{frame}



\begin{frame}{Empirical failure of the PIH (from EC1B1 notes)}
\vspace{-4mm}
\begin{figure}[tph]
\centering
\includegraphics[width=\textwidth]{figures/EC1B1_Johnson_etal.pdf} 
\end{figure}
\end{frame}


\begin{frame}{Empirical failure of the PIH (from EC1B1 notes)}
\vspace{-4mm}
\begin{figure}[tph]
\centering
\includegraphics[width=\textwidth]{figures/EC1B1_Johnson_etal2.pdf} 
\end{figure}
\end{frame}



\begin{frame}{Empirical failure of the PIH}
\vspace{-4mm}
\begin{figure}[tph]
\centering
\includegraphics[width=.67\textwidth]{figures/Ganong_etal_fig6.pdf} 
\end{figure}
Source: Ganong-Jones-Noel-Farrell-Greig-Wheat
\end{frame}



\begin{frame}{How to fix the PIH prediction?}
Huge literature. Most common fixes:
\bi
\medskip
\item Credit constraints, i.e. drop assumption that can borrow as much as you want at rate $r$
\medskip
\item Behavioral theories
\bi
\medskip
\item mental accounting
\medskip
\item present bias or ``salience of the present" $\Rightarrow$ dynamic inconsistency and procrastination, see EC1P1 lecture notes
\medskip
\item ...
\ei
\ei
\end{frame}


\begin{frame}{This is exactly what the HANK literature is all about}
\vspace{-4mm}
\bi
\item Graph from Kaplan, Moll and Violante (2018)
\begin{figure}[tph]
\centering
\includegraphics[width=0.4\textwidth]{figures/mpc500_quarterly_function_av.eps}
\end{figure}
\item When PIH is fixed, something interesting happens: model behaviour resembles Keynesian cross again, in particular sizable multipliers
\bigskip
\item But important difference: micro founded model, makes precise predictions about behavior as well as inequality, can use it to think about welfare
\ei
\end{frame}



\begin{frame}
\renewcommand{\baselinestretch}{1}
\vspace{2.3cm}
\begin{center}
{\huge Consumption, saving and interest rates \\[1ex] in general equilibrium}
\end{center}
\end{frame}

\begin{frame}{Consumption, saving, interest rates in general equilibrium}
\bi
\item So far: exogenously given interest rate $r$
\bigskip
\item Now: equilibrium determination of $r$
\bigskip
\item Continue to work with two-period model
\bigskip
\item Economy without production, i.e. endowment economy (like in EC2A1)
\bigskip
\item Main application: representative household model
\bigskip
\item But start with environment with multiple households $i=1,...,I$ for reason that will become clear shortly
\ei
\end{frame}



\begin{frame}{Borrowing and saving in equilibrium}
\vspace{-3mm}
\bi
\item $I$ households $i=1,...,I$
\bigskip
\item Recall: $a_i>0$ = saving, $a_i<0$ = borrowing
\bigskip
\item Claim: in a closed system, e.g. a closed economy or this classroom
$$\sum_{i=1}^I a_i = 0 $$
\item Logic: there is a saver for every borrower, i.e. you can only borrow if someone takes the other side of this trade and lends to you
\pause
\bigskip
\item What if all households are alike, i.e. $a_i = a$ for all $i=1,...,I$? Then clearly
$$a_i = 0 \quad \mbox{all} \ i=1,...,I $$
i.e. in equilibrium noone will borrow and noone will save
\bigskip
\item Next: competitive equilibrium with rep household $\Rightarrow$ exactly what happens
\ei
\end{frame}


\begin{frame}{Competitive equilibrium with representative household}
\tr{\textbf{Definition:}} a competitive equilibrium are quantities $(c_1,c_2,a)$ and an interest rate $r$ such that
\be
\medskip
\item Utility maximization: taking as given $r$, the representative household chooses $(c_1,c_2,a)$ to solve
$$\max_{c_1,c_2,a} \ u(c_1) + \beta u(c_2) \quad \mbox{s.t.} \quad c_1 + a= y_1, \quad 
c_2= y_2 + (1+r)a$$
\item Profit maximization: none because no production
\smallskip
\item Market clearing: demand = supply for goods and credit market
\begin{align*}
\mbox{goods in period 1:} \qquad  c_1 &= y_1 \\[0.5ex]
\mbox{goods in period 2:} \qquad  c_2 &= y_2 \\[0.5ex]
\mbox{credit market:} \qquad a &= 0
\end{align*}
\ee
Note on goods market: if no borrowing, can only consume your income
\end{frame}


\begin{frame}{What's the point of this equilibrium model?}
\bi
\item We usually solve for equilibrium quantities like $(c_1,c_2,a)$ because we're interested in their behavior, how they change with parameters etc
\bigskip
\item But here we already know that $(c_1,c_2,a) = (y_1,y_2,0)$ so what's even the point?
\bigskip
\item Answer: the model makes interesting predictions about the equilibrium interest rate
\ei
\end{frame}



\begin{frame}{Solving for the equilibrium interest rate}
\bi
\item Household maximization $\Rightarrow$ Euler equation
$$u'(c_1) = \beta(1 + r) u'(c_2) $$
\item But we know that $c_1 = y_1$ and $c_2 = y_2$. Therefore
$$1 + r^* = \frac{1}{\beta}\frac{u'(y_1)}{u'(y_2)} $$
\item Parametric example from earlier: $u'(c) = c^{-1/\sigma}$
$$1 + r^* = \frac{1}{\beta}\left(\frac{y_2}{y_1}\right)^{1/\sigma}  $$
\item Note: for entire economy, reasonable to assume $y_1 \approx y_2 \Rightarrow 1+r^* \approx 1/\beta$
\ei
\end{frame}


\begin{frame}{Comparative statics: how does $r^*$ depend on parameters?}
\bi
\item Recall
$$1 + r^* = \frac{1}{\beta}\left(\frac{y_2}{y_1}\right)^{1/\sigma}  $$
\item What happens to equilibrium interest rate $r^*$ when
\bi
\medskip
\item $\beta \uparrow$?
\medskip
\item $y_1 \uparrow$?
\medskip
\item $y_2 \uparrow$?
\ei
\medskip
and what is the intuition?
\bigskip
\item Note: questions like these are excellent exam questions!
\ei
\end{frame}



\begin{frame}{Next few lectures}
\bi
\item More interesting dynamic macro models with non-degenerate predictions for the quantities?
\bigskip
\item $\Rightarrow$ add production and capital accumulation
\bigskip
\item Use them to think about business cycles and macroeconomic stabilization policies, in particular monetary and fiscal policy
\bi
\medskip
\item microfounded analogue of Keynesian IS-MP-PC model from EC1B1
\ei
\ei
\end{frame}







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\end{document}

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