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\title{\vspace{1.6cm} \\
	{\LARGE Lecture 5}\\[1ex]
{\LARGE Investment and Capital Accumulation}\\
	\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}

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{
\setbeamertemplate{footline}{} 
\begin{frame}[noframenumbering]
\titlepage
\end{frame}
}


\begin{frame}{Investment over the business cycle}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.85\textwidth]{figures/jones_investment.pdf} 
\end{figure}
\vspace{-3mm}
Source: Jones, Macroeconomics
\end{frame}




\begin{frame}{Plan}
\be
\item Investment and capital accumulation in partial equilibrium
\bigskip
\item Dynamic general equilibrium with capital accumulation
\bigskip
\item Half-way mark for this course: taking stock and looking ahead
\ee
\end{frame}


\begin{frame}{Readings and supplementary materials}
\be
\bigskip
\item Supplement on moodle: write-up of model including all the derivations
\bi
\bigskip
\item I will provide such write-ups going forward, partly so we can skip the derivations during lectures
\ei
\bigskip
\item Chapters 8 and 9 of Kurlat
\bi
\medskip
\item general equilibrium model in his chapter 9 is similar but more complicated
\medskip
\item also includes labor demand and supply which we abstract from
\ei
\ee
\end{frame}


\begin{frame}
\renewcommand{\baselinestretch}{1}
\vspace{2.3cm}
\begin{center}
{\huge Investment and capital accumulation \\[1ex] in partial equilibrium}
\end{center}
\end{frame}


\begin{frame}{Investment and capital accumulation in partial equilibrium}
\bi
\item Two time periods $t=1,2$, world ends afterwards
\bigskip
\item Representative firm
\bigskip
\item Firms produce using capital $K_t,t=1,2$
$$Y_t = A_t F(K_t), \quad F'>0, F''<0$$
\item $K_1$ is fixed
\medskip
\item $K_2$ can be changed through investment in period $1$, $I_1$
$$K_2 = I_1 + (1-d)K_1, \quad 0 < d \leq 1 $$
where $d$ is depreciation (same equation as in Solow model)
\ei
\end{frame}



\begin{frame}{Firm profit maximization}
\vspace{-3mm}
\bi
\item Firm profits per period
$$\Pi_1 = A_1 F(K_1) - I_1, \quad \Pi_2 = A_2 F(K_2) + (1-d)K_2$$
\item Why $(1-d)K_2$ in $\Pi_2$?
\bigskip
\pause
\item Firms maximize present discounted value (PDV) of profits
$$W = \Pi_1 + \frac{\Pi_2}{1+r_1} $$
where $r_1$ is interest rate between $t=1$ and $t=2$
\bigskip
\pause
\item Why would firms maximize this present value? Why discounted at $r_1$?
\medskip
\item See supplement. In a nutshell because
\bi
\smallskip
\item[(a)] firms are owned by households who can save and borrow at $r_1$
\smallskip
\item[(b)] also firms can save and borrow at $r_1$
\ei
\smallskip
(a)+(b) $\Rightarrow$ households instruct firms to maximize $W = \Pi_1 + \frac{\Pi_2}{1+r_1}$
\ei
\end{frame}




\begin{frame}{Firm profit maximization}
\bi
\item Combining, firm problem is
$$W = \max_{\tr{K_2}} \ \left\{A_1 F(K_1) + (1-d)K_1 - \tr{K_2} + \frac{A_2 F(\tr{K_2}) + (1-d)\tr{K_2}}{1+r_1} \right\}$$
\bigskip
\item Optimality condition
$$1 = \frac{A_2 F'(K_2) + 1 - d}{1+r_1}$$
or
\tr{
\begin{equation}\tag{$\ast$}\label{eq:star}
A_2 F'(K_2) = r_1 + d 
\end{equation}}
\item \eqref{eq:star} pins down optimal capital $K_2^*$ and therefore optimal investment
$$I_1^* = K_2^* - (1-d)K_1 $$
\ei
\end{frame}


\begin{frame}{Graphical representation of optimality condition}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.78\textwidth]{figures/lec5_1.eps} 
\end{figure}
\end{frame}


\begin{frame}{How does investment respond to changes in $A_2$ and $r_1$?}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.49\textwidth]{figures/lec5_2.eps} \includegraphics[width=.49\textwidth]{figures/lec5_3.eps} 
\end{figure}
\bi
\item What is the intuition for these comparative statics?
\ei
\end{frame}





\begin{frame}{Capital and investment demand: parametric example}
\bi
\item Example: $F(K) = K^\alpha$ with $0<\alpha < 1$
\bigskip
\item Optimality condition
$$\alpha A_2 K_2^{\alpha-1} = r_1 + d $$
\item $\Rightarrow$ optimal capital demand and investment
\begin{align*}
K_2^* &= \left(\frac{\alpha A_2}{r_1 + d}\right)^{\frac{1}{1-\alpha}} \\
I_1^* &= \left(\frac{\alpha A_2}{r_1 + d}\right)^{\frac{1}{1-\alpha}} - (1-d)K_1 
\end{align*}
\ei
\end{frame}



\begin{frame}{Warmup for part 2: equilibrium in the capital market}
\vspace{-3mm}
\bi
\item Optimality condition $A_2 F'(K_2) = r_1 + d$ traces out a capital demand curve
\medskip
\item Plot of $r_1 = A_2 F'(K_2) - d$ vs $K_2$ \tr{is} capital demand curve in $(K_2,r_1)$-space
\medskip
\item Warmup for part 2: can put this together w model of capital supply $K_2^S(r_1)$
\ei
\vspace{-2mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.6\textwidth]{figures/lec5_4.eps}
\end{figure}
\end{frame}


\begin{frame}{Warmup: equilibrium with infinitely elastic capital demand}
\vspace{-3mm}
\bi
\item Interesting special case: linear production \tr{$Y_2 = A_2 K_2$}
\medskip
\item Optimality: $A_2 \gtrless r_1 + d \Rightarrow$ capital demand \tr{infinitely elastic} at $r_1 = A_2 - d$
\medskip
\item Equilibrium interest rate \tr{$r_1 = A_2 - d$}, quantity pinned down by supply
\ei
\vspace{-2mm}

\begin{figure}[tph]
	\centering
\includegraphics[width=.6\textwidth]{figures/lec5_5.eps}
\end{figure}
\end{frame}




\begin{frame}
\renewcommand{\baselinestretch}{1}
\vspace{2.3cm}
\begin{center}
{\huge Dynamic general equilibrium \\[1ex] with capital accumulation}
\end{center}
\end{frame}





\begin{frame}{Overview}
\bi
\item Lecture 4: household saving
\bigskip
\item Lecture 5 so far: firm capital accumulation and production
\bigskip
\item Now put these together in general equilibrium
\bigskip
\item To make progress, work with special functional forms
\bi
\medskip
\item no labor, linear production: $Y_t = A_t K_t$
\medskip
\item full depreciation $d=1$
\medskip
\item utility with constant intertemporal elasticity of substitution
\ei
\bigskip
\item Later: heavy use of these in our version of \tr{New Keynesian model}
\bigskip
\item Special assumptions but should be clear: construction of equilibrium follows steps that would also be valid with more general functions
\ei
\end{frame}




\begin{frame}{Primitives of economy with capital accumulation}
\bi
\item \tr{\textbf{Preferences:}} households have utility function
$$U(C_1) + \beta U(C_2) \quad \mbox{with} \quad U(C) = \frac{C^{1-\frac{1}{\sigma}}-1}{1-\frac{1}{\sigma}}$$
\item \tr{\textbf{Technology:}} firms have production function
$$Y_t = A_t K_t, \quad t=1,2 $$
and capital accumulates according to $K_2 = I_1 + (1-d)K_1$ with $d=1$, i.e.
$$K_2 = I_1 $$
\item \tr{\textbf{Resource constraints (feasibility):}}
\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*}
\ei
\end{frame}


\begin{frame}{Competitive equilibrium with capital accumulation}
\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}{Comment on market clearing conditions: credit market}
\vspace{-2mm}
\bi
\item There really is a third market and correponding market clearing condition
\bi
\medskip
\item credit market in which households and firms borrow/lend from/to each other at interest rate $r_1$ (recall that's why firms maximize PDV)
\medskip
\item but can drop this due to Walras' Law
\ei
\bigskip
\pause
\item Can write this as 
$$b + B = 0 $$
where
\bi
\medskip
\item $b$: household saving with $b<0$ denoting borrowing
\medskip
\item $B$: firm saving with $B<0$ denoting borrowing
\ei
\medskip
\item Typical situation $b>0$ and $B = - b<0$, i.e. firms borrow from households to finance their investment
\medskip
\item See supplement. Can forget about this from now, no future appearances.
\ei
\end{frame}


\begin{frame}{Solving for the competitive equilibrium allocation}
\bi
\bigskip
\item \tr{What is a convenient strategy to solve for the equilibrium allocation?}
\bigskip
\item Supplement: alternative strategy
\ei
\end{frame}



\begin{frame}{Result: competitive equilibrium allocation (see supplement)}
\vspace{-9mm}
\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*}
Derivation: see supplement (one of the derivations, there are at least two)
\end{frame}


\begin{frame}{Investment in the competitive equilibrium}
\vspace{-3mm}
\bi
\item Focus on investment which is most interesting decision
$$ I_1 = \frac{1}{1+\left(\frac{1}{\beta A_2} \right)^\sigma A_2}A_1 K_1 $$
\item Note: can write this as saving rate $s(A_2)$ out of current output, $Y_1$:
$$I_1 = s(A_2)Y_1, \quad C_1 = [1-s(A_2)]Y_1, \quad s(A_2) =\frac{1}{1+\left(\frac{1}{\beta A_2} \right)^\sigma A_2} $$
\item \tr{Contrast this with Solow model:} $I_t = s Y_t$ where $s$ = exogenously given
\bigskip
\item In current model, saving rate is instead endogenous and depends on preferences $(\beta,\sigma)$ and technology ($A_2$)!
\bigskip
\item This is precisely what we mean when we say ``Solow model is not microfounded but modern macro models are"
\ei
\end{frame}


\begin{frame}{A recession due to a drop in $A_1$}
\vspace{-3mm}
\bi
\item Assume current productivity \tr{$A_1 \downarrow$}. What happens to $(C_1,C_2,I_1,Y_1,Y_2)$?
\bi
\medskip
\item momentarily: what on earth is a drop in productivity? 
\ei
\bigskip
\item Recall expressions for $(C_1,C_2,I_1,Y_1,Y_2)$ a few slides ago, e.g.
$$
Y_1 = \tr{A_1} K_1, \qquad Y_2 = \frac{A_2}{1+\left(\frac{1}{\beta A_2} \right)^\sigma A_2}\tr{A_1} K_1, \quad ...$$
\item Clearly all of $(C_1,C_2,I_1,Y_1,Y_2)$ fall when \tr{$A_1$} falls
\bigskip
\item Intuition: economy is less productive, people are poorer $\Rightarrow$ cut consumption in both periods as well as investment
\bigskip
\item Also note: both $C_1$ and $I_1$ fall. Reason: total resources $Y_1$ available for consuming or investing fall
$$C_1 + I_1 = Y_1, \quad Y_1 = \tr{A_1} K_1 $$
\ei
\end{frame}



\begin{frame}{A recession due to a drop in $A_2$}
\vspace{-2mm}
\bi
\item Assume future productivity $A_2 \downarrow$. What happens to $(C_1,C_2,I_1,Y_1,Y_2)$?
\medskip
\item \tr{$Y_2$ unambiguously decreases when $A_2$ falls}
$$Y_2 = A_2 K_2 = \frac{1}{\frac{1}{\tr{A_2}}+\left(\frac{1}{\beta \tr{A_2}} \right)^\sigma}A_1 K_1 \downarrow \quad \mbox{when} \quad \tr{A_2 \downarrow}.$$
\pause
\vspace{-3mm}
\item But: effect on $I_1$ and $C_1$ is \tr{ambiguous}, one falls the other rises
$$I_1 = s(\tr{A_2})Y_1, \quad C_1 = [1-s(\tr{A_2})]Y_1, \quad s(\tr{A_2}) =\frac{1}{1+\left(\frac{1}{\beta \tr{A_2}} \right)^\sigma \tr{A_2}} $$
\vspace{-4mm}
\item What happens to $I_1$ and $C_1$ depends on $\sigma$ = IES
$$\sigma<1 \quad \Rightarrow \quad s'(A_2)<0 \quad \Rightarrow \frac{\partial I_1}{\partial A_2} < 0, \quad  \frac{\partial C_1}{\partial A_2} > 0 \quad \Rightarrow \quad I_1 \uparrow, \ C_1 \downarrow $$
\vspace{-4mm}
\item Why? And what is the intuition?
\medskip
\item From now on always assume $\sigma<1$ (= also the empirically realistic case)
\ei
\end{frame}


\begin{frame}{A recession due to a drop in both $A_1$ and $A_2$}
Can also consider combination: both $A_1$ and $A_2$ fall at same time
\bi
\medskip
\item For example $A_1$ falls and $A_2$ is correlated with $A_1$
$$\log A_2 = \rho \log A_1 + \varepsilon_2$$
where $\rho$ captures persistence, $\varepsilon_2$ the innovation at $t=2$
\bigskip
\item Economic effect is combination of effects on two previous slides
\bigskip
\item Such correlated productivity movements will be important in next lecture
\ei
\end{frame}



\begin{frame}{Oil shocks as productivity shocks (or gas shocks)}
\vspace{-3mm}
\bi
\item What on earth is a drop in productivity? 
\bi
\medskip
\item do we think people forget how to make stuff? Not really
\medskip
\item hard to come up with sensible justifications
\ei
\bigskip
\item One possible justification: oil shocks or energy (gas etc) shocks
\pause
\bigskip
\item Technology: firms use oil to produce
$$\tilde{Y}_t =  \tilde{A}_t K_t^\alpha O_t^{1-\alpha}.$$
\item Firms maximize output net of oil expenditure
$$Y_t = \max_{O_t} \ \tilde{A}_t K_t^\alpha O_t^{1-\alpha} - p_t O_t \quad \mbox{where} \ p_t = \mbox{oil price} $$
$$\Rightarrow \quad Y_t = A_t K_t \ \ \mbox{where} \ \ A_t = \mbox{effective productivity} = \alpha \tilde{A}_t^{1/\alpha} \left(\frac{1-\alpha}{p_t} \right)^{(1-\alpha)/\alpha}  $$
so an increase in $p_t$ causes a drop in effective productivity
\ei
\end{frame}






\begin{frame}
\renewcommand{\baselinestretch}{1}
\vspace{2.3cm}
\begin{center}
{\huge Half-way mark for this course:\\[1ex] taking stock and looking ahead}
\end{center}
\end{frame}






\begin{frame}{What we have done in first half of course}
\vspace{-2mm}
\bi
\item Learned the basics of modern macroeconomics
\bigskip
\item What is a model? ``The map is not the territory" and so on
\bigskip
\item Reminded ourselves of basic microeconomics
\bi
\smallskip
\item \tr{intra}temporal choice, e.g. labor supply
\smallskip
\item \tr{inter}temporal choice, e.g. consumption, saving, investment
\ei
\bigskip
\item Learned the following key concepts
\be
\medskip
\item Competitive equilibrium
\medskip
\item Pareto effiency
\medskip
\item Welfare theorems
\ee
\ei
\end{frame}


\begin{frame}{What we have done in first half of course}
\bi
\item Worked with some key parameters, corresponding functional forms
\bi
\medskip
\item elasticity of substitution in production
\medskip
\item intertemporal elasticity of substitution
\ei
\bigskip
\item Reminded ourselves that income and substitution effects are everywhere!
\bigskip
\item Applied these tools to think about some key topics
\be
\bigskip
\item Labor supply and labor demand, long-run trends in hours worked
\bigskip
\item Substitution in production, e.g. in response to cut-off of Russian gas
\bigskip
\item Consumption and saving decisions, permanent income hypothesis \& its shortcomings, MPCs out of transitory \& permanent income shocks\\
\bigskip
\item Investment and capital accumulation
\ee
\ei
\end{frame}



\begin{frame}{What we will do in second half of course}
\bi
\vspace{-4mm}
\item Put your basic training to use to think about more and more applied topics
\medskip
\begin{figure}[tph]
	\centering
\includegraphics[width=.78\textwidth]{figures/topics_part2.pdf} 
\end{figure}
\item Looking for something to read during reading week? Read  Mankiw and Weinzierl (2011) ``An Exploration of Optimal Stabilization Policy" {\scriptsize \url{https://www.brookings.edu/wp-content/uploads/2011/03/2011a_bpea_mankiw.pdf}}
\bi
\smallskip
\item Two-period version of New Keynesian model I will teach is based on this paper
\smallskip
\item Model from this lecture but with ``nominal rigidities", i.e. sticky prices
\smallskip
\item You have basic tools to read this but may still lack some vocabulary
\smallskip
\item Read sections I-IV and IX-X (skip sections V-VIII)
\ei
\ei
\end{frame}






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\end{document}

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