\documentclass[10pt]{beamer}
%\documentclass[handout, 10pt]{beamer}

\mode<presentation>
\usetheme{Madrid}
%\definecolor{lightblue}{HTML}{3498DB}
\definecolor{lightblue}{HTML}{5DADE2}
\definecolor{foundersrock}{HTML}{3B7EA1}
\definecolor{goldenbear}{HTML}{FDB515}
\definecolor{berkeleyblue}{HTML}{003262}
%\definecolor{colblue2}{HTML}{0072CE} % Juan's blue
\usecolortheme[named=lightblue]{structure}
\setbeamercolor{button}{fg=white, bg=lightblue}

%\usecolortheme{seahorse}
%\usecolortheme{rose}
\usefonttheme[onlylarge]{structuresmallcapsserif}
%\usefonttheme[onlymath]{serif}

\definecolor{alertcolor}{rgb}{0.8,0,0} % dark red
%\definecolor{alertcolor}{HTML}{5DADE2} % light blue
\newcommand{\hl}[1]{\textcolor{darkred}{#1}}
\definecolor{beamer@darkred}{rgb}{0.8,0,0}
\setbeamercolor{alerted text}{fg=beamer@darkred}

\renewcommand{\baselinestretch}{1.3}   
\setbeamersize{text margin left=0.5cm}
\setbeamersize{text margin right=0.5cm}

\setbeamertemplate{itemize items}[boadilla]
\setbeamertemplate{enumerate items}[default]

\usenavigationsymbolstemplate{}

\renewcommand{\baselinestretch}{1.3}   
\setbeamersize{text margin left=0.5cm}
\setbeamersize{text margin right=0.5cm}

\usepackage[english]{babel}
\usepackage[latin1]{inputenc}
\usepackage{times}
\usepackage[T1]{fontenc}
\usepackage{epstopdf}
\usepackage{hyperref}
\usepackage{appendixnumberbeamer}

\usepackage{tikz}
\usepackage{pgfplots}
\usetikzlibrary{decorations.markings,decorations.pathmorphing,calc,intersections}

\title[Ideas and Growth]{Ideas and Economic Growth}
\author[Steinsson]{Jón Steinsson}
\institute[]{University of California, Berkeley}
\date[]{Fall 2025}

\begin{document}

\begin{frame}
  \titlepage
\end{frame}

\begin{frame}{Big Picture Questions about Growth}
	\begin{itemize}
		\itemsep1em 
		\item What sustains growth at the frontier? \\ {\small (Will it continue in the future?)}
		\item Why are some countries so far behind the frontier? \\ {\small (What might help them close the gap?)}
	\end{itemize}
	
	\vspace{30pt}
	This lecture focuses on the first of these questions
\end{frame}


\begin{frame}{Knowledge versus Capital}
	\begin{itemize}
		\itemsep1em 
		\item Solow model: Capital accumulation not a source of long-run growth
		\begin{itemize}
			\item Reason: Diminishing returns
		\end{itemize}
		\item What about knowledge?
		\item If knowledge succeeds where capital fails, there must be something fundamentally different about knowledge than capital
	\end{itemize}
\end{frame}


\begin{frame}{The AK Model}
	\begin{itemize}
		\item To drive home the importance of diminishing returns, let's consider \\ a model without diminishing returns
		\item Suppose 
		\[ Y(t) = A K(t) \]
		and 
		\[ \dot{K}(t) = s Y(t) - \delta K(t) \]
		where 
		\begin{itemize}
			\item $s$ is the exogenous savings rate (as in Solow model)
			\item Labor is assumed constant and normalized to one \\ (which implies that $Y(t)$ is output per person)
		\end{itemize}
	\end{itemize}
\end{frame}


\begin{frame}{The AK Model}
	\begin{itemize}
		\itemsep1em 
		\item Combining these two equations yields
		\[ \dot{Y}(t) = sA Y(t) - \delta Y(t) \]
		\[ g_Y = \frac{\dot{Y}_t}{Y_t} = s A - \delta \]
		\item We get long-run growth from capital accumulation
		\item The long-run growth rate of output (per person) \\ is governed by $s$, $A$, and $\delta$ 
		\item Long-run growth is endogenous to the extent that $s$, $A$, and $\delta$ \\ can be influenced by policy / behavior
	\end{itemize}
\end{frame}


\begin{frame}{Growth from Externalities}
	\begin{itemize}
		\itemsep1em 
		\item But why might we think $Y = AK$ makes sense?
		\item One ``micro-foundation'' is learning-by-doing externalities
		\begin{itemize}
			\item Productivity gains coming from investment and production
			\item Empirical evidence from airframe manufacturing, shipbuilding, etc. \\ (Wright 36, Searle 46, Asher 56, Rapping 65)
		\end{itemize}
		\item Several early endogenous growth models followed this path \\ {\small (e.g., Frankel 62, Griliches 79, Romer 86, Lucas 88)}
		\item We consider Romer (1986) version here \\ {\small (see Romer 19, p. 119-121; Barro and Sala-i-Martin 04, sec. 4.3; \\ Acemoglu 09, sec. 11.4)}
	\end{itemize}
\end{frame}


\begin{frame}{Romer (1986): Knowledge Spillovers}
	\begin{itemize}
		\item Suppose there is a continuum of firms with production function
		\[ Y_i(t) = F(K_i(t), A_i(t) L_i(t)) \]
		\item Two assumptions:
		\begin{itemize}
			\item Strong learning-by-doing (investing): \\ Knowledge grows proportionally with firm's capital stock
			\item Knowledge spillovers are perfect across firms \\ (all firms benefit from each firm's learning)
		\end{itemize}
		\item These assumptions imply:
		\[A_i(t) = B K(t) \]
	\end{itemize}
\end{frame}


\begin{frame}{Romer (1986): Knowledge Spillovers}
	\begin{itemize}
		\item Combining prior two equations:
		\[ Y_i(t) = F(K_i(t), B K(t) L_i(t)) \]
		\item Suppose further that all firms are identical:
		\[ Y(t) = F(K(t), B K(t) L(t)) \]
		\item If $F$ is homogeneous of degree one, we have
		\[ Y(t) = F(1,BL(t)) K(t) \]
		\item This model therefore yields a production function of the $Y = AK$ form
	\end{itemize}
\end{frame}


\begin{frame}{Growth and Knowledge Spillovers}
	\begin{itemize}
		\itemsep1em 
		\item Romer (1986) model yields endogenous growth
		\item But arguably makes unrealistic assumptions:
		\begin{itemize}
			\item Assumes very large amounts of learning-by-doing
			\item Doesn't work if knowledge grows less than proportionally with K
		\end{itemize}
		\item Lucas (1988) builds similar model with human capital externalities.
		\begin{itemize}
			\item Arguably also makes unrealistic assumptions \\ (see Jones 22, section 2.2)
		\end{itemize}
		\item Doesn't seem to capture what is ``special'' about knowledge
	\end{itemize}
\end{frame}


\begin{frame}{Why Is Knowledge Special?}
	\begin{itemize}
		\itemsep1em 
		\item Knowledge is non-rival
		\item This is the fundamental difference versus capital
		\item Implies that knowledge can be a source of long-run growth
	\end{itemize}
\end{frame}


\begin{frame}{Ideas vs. Objects}
	\begin{itemize}
		\itemsep1em 
		\item Ideas: a design, a blueprint, or a set of instructions
		\begin{itemize}
			\item How to make fire using sticks, calculus, the design of the incandescent light bulb, oral rehydration therapy, Beethoven's 3th symphony, etc.
		\end{itemize}		
		\item Objects: Goods, capital, labor, land, highways, barrels of oil, etc.
		\begin{itemize}
			\item A particular incandescent light bulb, a particular oral rehydration pill, etc. 
		\end{itemize} 
	\end{itemize}
\end{frame}


\begin{frame}{Ideas vs. Objects}
	\begin{itemize}
		\itemsep1em 
		\item Objects are rival:
		\begin{itemize}
			\item If I use a particular lawn mower, you can't use that same \\ lawn mower at the same time
		\end{itemize}
		\item Ideas are non-rival:
		\begin{itemize}
			\item My use of calculus, does not negatively affect your ability \\ to use calculus at the same time
			\item Once invented, calculus can be used by any number of people simultaneously (ideas are ``infinitely usable'')
		\end{itemize}
	\end{itemize}
\end{frame}

\begin{frame}{Non-Rivalry and Returns to Scale}
	\begin{itemize}
		\itemsep1em 
		\item Consider production function
		\[ Y = F(A,X) \] \vspace{-20pt}
		\begin{itemize}
			\item $A$ is index of the stock of knowledge
			\item $X$ is all rival inputs (vector) 
		\end{itemize} 
		\item Replication implies constant returns to objects:
		\[ \lambda Y = F(A,\lambda X) \] \vspace{-20pt}
		\begin{itemize}
			\item This argument implicitly uses non-rivalry of ideas
			\item We can use same A to build second factory as first factory.
		\end{itemize}
		\item Implies that if we increase A as well we get increasing returns:
		\[ F(\lambda A, \lambda X) > F(A, \lambda X) = \lambda Y \]
	\end{itemize}
\end{frame}


\begin{frame}{Non-Rivalry and Growth}
	\begin{itemize}
		\itemsep1em 
		\item Since ideas are non-rival, per capita output depends on the overall stock of knowledge, NOT knowledge \textbf{per capita}
		\[ Y(t) = A(t)^{\sigma} K(t)^{\alpha} L(t)^{1-\alpha}  \]
		\[ y(t) = A(t)^{\sigma} k(t)^{\alpha}\]
		\item Output per person depends on:
		\begin{itemize}
			\item Total stock of knowledge ($A(t)^{\sigma}$)
			\item Capital \textbf{per capita} ($k(t)^{\alpha}$)
		\end{itemize}
		\item Solow model: Capital \textbf{per capita} can't grow forever {\small (if $A$ is constant)}
		\item If stock of knowledge can grow forever, $y(t)$ can growth forever
	\end{itemize}
\end{frame}


\begin{frame}{Romer (1990)}
	\begin{itemize}
		\itemsep1em 
		\item Romer (1990) is the paper that crystallized these ideas 
		\item See Jones (2019) for role of this paper in relation to earlier and subsequent literature
		\item But Romer (1990) made some extreme assumptions that \\ we will want to move away from
	\end{itemize}
\end{frame}


\begin{frame}{Knowledge Production}
	\begin{itemize}
		\itemsep1em 
		\item Key new feature: Knowledge is produced
		\item Workers do one of two things:
		\begin{itemize}
			\item Produce goods and services
			\item Produce knowledge (R\&D)
		\end{itemize}
		\item Key choice: How are workers allocated between these activities?
		\item Simplifying assumption: A fraction $s$ of workers work on R\&D
		\begin{itemize}
			\item Similar to Solow assumption about savings rate
			\item Workers choose optimally in Romer (1990)
			\item We will consider a model where workers choose optimally later on
			\item For now:
		\end{itemize}
	
		\vspace{-20pt}
		\[ L_A(t) = sL(t) \hspace{40pt} L_Y(t) = (1-s) L(t)  \] 
	\end{itemize}
\end{frame}



\begin{frame}{Knowledge Production in Romer (1990)}
	\begin{itemize}
		\itemsep1em 
		\item Knowledge production function in Romer (1990):
		\[ \dot{A}(t) = \theta L_{A}(t) A(t) \]
		\item Knowledge production depends on two inputs:
		\begin{itemize}
			\item Research effort: $L_{A}(t)$ denotes labor devoted to research
			\item Existing knowledge: $A(t)$
		\end{itemize}
		\item Importantly, exponent on $A(t)$ is one
		\item Implies that 
		\[ g_{A}(t) = \frac{\dot{A}(t)}{A(t)} = \theta L_{A}(t) \] 
	\end{itemize}
\end{frame}


\begin{frame}{Knowledge Production in Romer (1990)}
	\begin{itemize}
		\itemsep1em 
		\item Suppose for simplicity that $L_{A}(t) = L_{A}$ (i.e., a constant)
		\item Then growth rate of knowledge is constant
		\[ g_{A} = \frac{\dot{A}(t)}{A(t)} = \theta L_{A} \]
		\item Suppose for simplicity that goods production function is
		\[ Y(t) = A(t)^{\sigma} L_{Y} \hspace{20pt} => \hspace{20pt} y(t) = A(t)^{\sigma} (1-s) \]
		where $1-s$ is (constant) share of pop. working on goods production, \\ $\sigma$ is importance of ideas for production (degree of increasing returns)
		\item This implies 
		\[ g_y = \sigma g_A = \sigma \theta L_A \]  
	\end{itemize}
\end{frame}

\begin{frame}{Knowledge Production}
	\begin{itemize}
		\itemsep1em 
		\item But why would knowledge production be linear in A(t) and L(t)?
		\item More generally:
		\[\dot{A}(t) = \theta L_{A}(t)^{\lambda} A(t)^{\phi} \]
		\item Not necessarily constant returns to objects ($\lambda = 1$):
		\begin{itemize}
			\item Twice as much research effort may not generate twice as much knowledge
			\item There may be congestion / duplication / diminishing returns
			\item This would yield $\lambda < 1$
			\item We assume however that $\lambda > 0$
		\end{itemize} 
	\end{itemize}
\end{frame}


\begin{frame}{Knowledge Production}
	\[\dot{A}(t) = \theta L_{A}(t)^{\lambda} A(t)^{\phi} \] \vspace{-15pt}
	\begin{itemize}
		\itemsep1em 
		\item Not necessarily constant returns to existing knowledge ($\phi = 1$)
		\item $\phi > 0$: Standing on the shoulders of giants
		\begin{itemize}
			\item Having more knowledge lets a researcher create knowledge faster
			\item E.g., printed books, internet, computers, microscopes, etc.
		\end{itemize}
		\item $\phi < 0$: No more low hanging fruit
		\begin{itemize}
			\item Suppose you are fishing in a pond with 100 fish
			\item As you catch more, harder to catch the rest
		\end{itemize} 
		\item Nothing particularly natural about $\phi = 1$
	\end{itemize}
\end{frame}


\begin{frame}{Simple Endogenous Growth Model}
	\begin{equation*}
		\begin{array}{ll}
			\text{1. Goods production:} & Y(t) = A(t)^{\sigma} L_{Y}(t) \\ [2ex]
			\text{2. Ideas production:} & \dot{A}(t) = \theta L_A(t)^{\lambda} A(t)^{\phi} \\ [2ex]
			\text{3. Allocation:} & L_A(t) = s L(t) \\ [2ex]
			\text{4. Resource constraint:} & L(t) = L_A(t) + L_Y(t) \\ [2ex]
			\text{5. Population growth:} & L(t) = L(0) e^{nt}
		\end{array}
	\end{equation*}
\end{frame}


\begin{frame}{Simple Endogenous Growth Model}
	Notable features: \vspace{5pt}
	\begin{itemize}
		\itemsep1em 
		\item Constant fraction of labor force $s$ conducts research
		\begin{itemize}
			\item Simple short cut
			\item Similar to constant savings rate in Solow model
			\item We will endogenize later
		\end{itemize}
		\item Constant population growth at rate $n$
		\item $\sigma$ captures degree to which increase in knowledge \\ increases productivity in production of goods and services
		\begin{itemize}
			\item How much does 1\% increase in knowledge increase productivity?
			\item But what is a 1\% increase in knowledge? How is this measured? 
		\end{itemize}
	\end{itemize}
\end{frame}


\begin{frame}{Balanced Growth in Simple Model}
	\begin{itemize}
		\item Combining (1), (3) and (4) and dividing by L(t) we get:
		\[y(t) = A(t)^{\sigma}(1-s) \]
		\item Taking logs and time derivatives yields
		\[g_y(t) = \sigma g_A(t) \]
		\item Suppose there is a balanced growth path with constant growth: 
		\[ g_y(t) = g_y  \hspace{20pt} \text{and} \hspace{20pt} g_A(t) = g_A \] 
		\item Then we have
		\[ g_y = \sigma g_A \] 
	\end{itemize}
\end{frame}


\begin{frame}{Balanced Growth in Simple Model}
	\begin{itemize}
		\item Combining (2) and (3) and dividing by $A(t)$:
		\[g_A(t) = \theta s^{\lambda} L(t)^{\lambda} A(t)^{\phi-1} \]
		\item Taking logs and time derivatives yields
		\[ 0 = \lambda g_L + (\phi-1) g_A \]
		where we use $g_A(t) = g_A$ on BGP
		\item Rearranging and using $g_L = n$ we get
		\[ g_y = \sigma g_A = \frac{\sigma \lambda}{1-\phi} n \]     
	\end{itemize}
\end{frame}


\begin{frame}{Output Growth and Population Growth}
	\[ g_y = \sigma g_A = \frac{\sigma \lambda}{1-\phi} n \] \vspace{-10pt}
	\begin{itemize}
		\item Long-run growth proportional to population growth rate
		\item If $L_A(t)$ were constant at $L_A$ (which implies $n=0$):
		\[ \frac{\dot{A}(t)}{A(t)} = \theta L_A^{\lambda} A(t)^{\phi-1} = \frac{\theta L_A^{\lambda}}{A(t)^{1-\phi}} \]
		\item If $1 - \phi > 0$, or equivalently $\phi < 1$:
		\[ g_A(t) = \frac{\dot{A}(t)}{A(t)} \rightarrow 0\]
		\item Growth can't keep up with the level and thus goes to zero 
	\end{itemize}
\end{frame}


\begin{frame}{Research Effort Must Grow Exponentially }
	\[ g_y = \sigma g_A = \frac{\sigma \lambda}{1-\phi} n \] \vspace{-10pt}
	\begin{itemize}
		\itemsep1em 
		\item With $\phi < 1$, research effort must grow exponentially for \\ knowledge to grow exponentially
		\item Exponential population growth and constant share of labor force working on research ($s$) does the trick
	\end{itemize}
\end{frame}


\begin{frame}{Something Must Have Linear Differential Eq.}
	Three ways to get sustained growth:
	\begin{enumerate}
		\itemsep1em 
		\item AK Model: Capital accumulation linear differential eq.
		\[ \dot{K}(t) = s A K(t) - \delta K(t) \hspace{20pt} => \hspace{20pt} \dot{K}(t) = (s A - \delta) K(t) \] 
		\item Romer (1990) / $\phi = 1$: Knowledge prod. linear differential eq. 
		\[ \dot{A}(t) = \theta L_{A}(t) A(t) \] \vspace{-20pt}
		\begin{itemize}
			\item ``Fully-endogenous'' growth model
			\item Also true of Aghion-Howitt 92, Grossman-Helpman 91
		\end{itemize}
		\item Jones (1995) / $\phi < 1$: Pop. growth linear differential eq. 
		\[ \dot{A}(t) = \theta L_{A}(t) A(t)^{\phi} \hspace{40pt} \dot{L}(t) = n L(t) \] \vspace{-20pt}
		\begin{itemize}
			\item ``Semi-endogenous'' growth model
		\end{itemize}
	\end{enumerate}
\end{frame}


\begin{frame}{Evolution of Growth in Simple Model}
	\begin{itemize}
		\item Growth of knowledge is generally (even outside BGP):
		\[g_A(t) = \theta s^{\lambda} L(t)^{\lambda} A(t)^{\phi-1} \]
		\item Taking logs and differentiating by time yields
		\[ \frac{\dot{g}_{A}(t)}{g_A(t)} = \lambda n - (1-\phi) g_A(t) \]
		\item Multiplying through by $g_A(t)$ yields
		\[ \dot{g}_{A}(t) = \lambda n g_A(t) - (1-\phi) g_A(t)^2 \]  
	\end{itemize}
\end{frame}


\begin{frame}{Evolution of Growth in Simple Model}
	\begin{equation} \label{gAlevel}
		g_A(t) = \theta s^{\lambda} L(t)^{\lambda} A(t)^{\phi-1}
	\end{equation} \vspace{-10pt}
	\begin{equation} \label{gAchange}
		\dot{g}_{A}(t) = \lambda n g_A(t) - (1-\phi) g_A(t)^2
	\end{equation} \vspace{-10pt}
	\begin{itemize}
		\itemsep1em 
		\item Equation \eqref{gAlevel} determines initial level of $g_A(t)$
		\begin{itemize}
			\item Depends, e.g., on $s$ (and therefore innovation policy)
		\end{itemize}
		\item Equation \eqref{gAchange} determines subsequent evolution of $g_A(t)$
		\begin{itemize}
			\item Independent of $s$
		\end{itemize}
		\item With $\phi < 1$ a change in $s$ only has a ``level effect'', not a ``growth effect''
	\end{itemize}
\end{frame}


\begin{frame}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/Romer2019Figure31.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Romer (2019). In Romer's notation $\theta < 1$ is what I have called $\phi < 1$.}
\end{frame}


\begin{frame}
	\begin{figure}
		\centering
		\includegraphics[height = 0.85  \textheight]{FiguresTables/Romer2019Figure32.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Romer (2019). In Romer's notation $\theta < 1$ is what I have called $\phi < 1$ and $a_L$ is what I have called $s$}
\end{frame}


\begin{frame}
	\begin{figure}
		\centering
		\includegraphics[height = 0.85  \textheight]{FiguresTables/Romer2019Figure33.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Romer (2019). In Romer's notation $\theta < 1$ is what I have called $\phi < 1$ and $a_L$ is what I have called $s$}
\end{frame}


\begin{frame}{Effect of $s$ on Growth}
	\[  g_{A}(t) = \frac{\dot{A}(t)}{A(t)} = \theta L_A^{\lambda} A(t)^{\phi-1} = \frac{\theta s^\lambda L(t)^{\lambda}}{A(t)^{1-\phi}} \]
	\begin{itemize}
		\item Models with $\phi = 1$: $s$ affects long run growth rate
		\[  g_{A}(t) = \frac{\dot{A}(t)}{A(t)} = \theta s^\lambda L(t)^{\lambda} \]
		\item Models with $\phi < 1$: $s$ does not affect long run growth rate
		\[ g_y = \sigma g_A = \frac{\sigma \lambda}{1-\phi} n \]
	\end{itemize}
\end{frame}


\begin{frame}{Scale Effects}
	\begin{itemize}
		\itemsep1em 
		\item Models with $\phi = 1$ have ``strong'' scale effects
		\begin{itemize}
			\item Growth rate is increasing in \textbf{level} of population:
			\[ g_{A}(t) = \frac{\dot{A}(t)}{A(t)} = \theta s^\lambda L(t)^{\lambda} \]
		\end{itemize}
		\item Models with $\phi < 1$ have ``weak'' scale effects
		\begin{itemize}
			\item Growth rate is increasing in \textbf{growth rate} of population:
			\[ g_y = \sigma g_A = \frac{\sigma \lambda}{1-\phi} n \]
		\end{itemize}
		\item These are interesting testable implications of these model classes
	\end{itemize}
\end{frame}


\begin{frame}{Do Scale Effects Apply at Country Level?}
	\begin{itemize}
		\itemsep1em 
		\item One reading of scale effects is that large countries or countries with \\ fast population growth should have high TFP growth
		\item Obviously counterfactual (Luxembourg, Iceland, Singapore)
		\item But ideas flow between countries
		\item Scale effects likely to operate largely at the world level \\ (although flow of ideas is not perfect or instantaneous)
	\end{itemize}
\end{frame}


\begin{frame}{Strong Scale Effects}
	\begin{itemize}
		\itemsep1em 
		\item There is arguably very strong evidence against strong scale effects: \vspace{5pt}
		\begin{itemize}
			\itemsep1em 
			\item Frontier growth has been quite stable for a long time
			\item Research effort has increased very substantially 
		\end{itemize}
		\item With strong scale effects, increased research effort should \\ increase TFP growth at frontier
	\end{itemize}
\end{frame}


\begin{frame}
	\begin{figure}
		\centering
		\includegraphics[height = 0.85  \textheight]{FiguresTables/Jones1995bFigure1.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Jones (1995).}
\end{frame}


\begin{frame}
	\begin{figure}
		\centering
		\includegraphics[height = 0.9  \textheight]{FiguresTables/Jones1995bFigure4.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Jones (1995).}
\end{frame}


\begin{frame}
	\begin{figure}
		\centering
		\includegraphics[height = 0.9  \textheight]{FiguresTables/Jones1995bFigure5.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Jones (1995).}
\end{frame}


\begin{frame}
	\begin{figure}
		\centering
		\includegraphics[height = 0.88  \textheight]{FiguresTables/Jones2005Figure2.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Jones (2005).}
\end{frame}


\begin{frame}{Evidence Against Strong Scale Effects}
	\[ g_{A}(t) = \frac{\dot{A}(t)}{A(t)} = \theta s L(t) \] \vspace{-10pt}
	\begin{itemize}
		\itemsep1em 
		\item Research effort has risen by a factor of 8
		\item Models with $\phi = 1$ imply that growth should have \\ increased by a factor of 8
		\item Clearly way off!
	\end{itemize}
\end{frame}


\begin{frame}{Ideas Harder to Find}
	\begin{itemize}
		\itemsep1em 
		\item This evidence suggests that ideas are harder to find
		\item By ideas, we mean ``proportional increases in productivity''
		\item Research productivity is falling. It takes more research effort \\ to produce the same growth rate
		\item This means $\phi < 1$ ($\beta > 0$ using Jones (2022) notation)
		\item But by how much?
		\begin{itemize}
			\item If $\phi = 0.95$ growth effects of change in $s$ on transition path \\ would last for a long time
		\end{itemize}
	\end{itemize}
\end{frame}


\begin{frame}{Bloom, Jones, Van Reenen, Webb (2020)}
	\begin{itemize}
		\itemsep0.5em 
		\item Estimate extent to which ideas are getting harder to find \\ at both macro and micro level
		\item Ideas production function
		\[ \frac{\dot{A}(t)}{A(t)} = \alpha A(t)^{-\beta} S(t) \] \vspace{-15pt}
		\begin{itemize}
			\item $S(t)$ denotes ``scientists'' (i.e., research effort in units of people)
			\item Notice that $\beta = 1 - \phi$
		\end{itemize}
		\item If $g_A$ is constant:
		\[ \beta = \frac{g_S}{g_A} \]
		\item Define:
		\[ \text{Research Productivity} = \frac{\dot{A}(t)/A(t)}{S(t)} \] 
	\end{itemize}
\end{frame}


\begin{frame}{Bloom, Jones, Van Reenen, Webb, (2020)}
	\begin{itemize}
		\itemsep0.5em 
		\item Consider the ``lab equipment'' model of research
		\[\dot{A}_t = \alpha R_t \]
		\item $R_t$ denotes research expenditures (labor and capital/equipment)
		\item Divide through by $A_t$ and multiply-divide by $w_t$ (wage):
		\[\frac{\dot{A}_t}{A_t} = \underbrace{\frac{\alpha w_t}{A_t}}_{\text{Research Prod.}} \times  \underbrace{\frac{R_t}{w_t}}_{\text{Scientists}} \] 
		\item In Romer-type model $R_t/w_t$ and $w_t/A_t$ are constant 
		\item Measure research effort (in units of ``scientists'') by $R_t/w_t$
	\end{itemize}
\end{frame}


\begin{frame}{Aggregate Evidence}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75 \textheight]{FiguresTables/BloomJonesVanReenenWebb2020Figure1.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Bloom, Jones, Van Reenen, Webb (2020).}
\end{frame}


\begin{frame}{Moore's Law}
	\begin{figure}
		\centering
		\includegraphics[height = 0.8 \textheight]{FiguresTables/BloomJonesVanReenenWebb2020Figure3.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Bloom, Jones, Van Reenen, Webb (2020).}
\end{frame}


\begin{frame}{Moore's Law}
	\begin{figure}
		\centering
		\includegraphics[height = 0.8 \textheight]{FiguresTables/BloomJonesVanReenenWebb2020Figure4.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Bloom, Jones, Van Reenen, Webb (2020). Research expenditure for several dozen specific firms.}
\end{frame}


\begin{frame}{Moore's Law}
	\begin{figure}
		\centering
		\includegraphics[height = 0.8 \textheight]{FiguresTables/BloomJonesVanReenenWebb2020Figure6.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Bloom, Jones, Van Reenen, Webb (2020). Productivity is yield per acre. }
\end{frame}


\begin{frame}
	\begin{figure}
		\centering
		\includegraphics[height = 0.95 \textheight]{FiguresTables/BloomJonesVanReenenWebb2020Table7.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Bloom, Jones, Van Reenen, Webb (2020).}
\end{frame}


\begin{frame}{Measuring Innovation Using Patents}
	\begin{itemize}
		\itemsep0.5em 
		\item Large literature that uses numbers of patents to measure innovation
		\item But most patents are not very innovative
		\item Refinements:
		\begin{itemize}
			\item Patent citations
			\item Market value of patents
		\end{itemize}
		\item Kelly, Papanikolaou, Seru, Taddy (2021):
		\begin{itemize}
			\item Use text similarity of patents
			\item Important patents are different from previous patents (novel) and \\ similar to subsequent patents (impactful)
		\end{itemize} 
	\end{itemize}
\end{frame}


\begin{frame}
	\begin{figure}
		\centering
		\includegraphics[height = 0.95 \textheight]{FiguresTables/KellyPapanikolaouSeruTaddy2021Figure2.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Kelly, Papanikolaou, Seru, Taddy (2022).}
\end{frame}


\begin{frame}
	\begin{figure}
		\centering
		\includegraphics[height = 0.95 \textheight]{FiguresTables/KellyPapanikolaouSeruTaddy2021Figure4a.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Kelly, Papanikolaou, Seru, Taddy (2022).}
\end{frame}


\begin{frame}
	\begin{figure}
		\centering
		\includegraphics[height = 0.95 \textheight]{FiguresTables/KellyPapanikolaouSeruTaddy2021Figure5.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Kelly, Papanikolaou, Seru, Taddy (2022).}
\end{frame}

\begin{frame}{Growth in the Past and Future}
	\begin{itemize}
		\itemsep1em 
		\item Semi-endogenous growth model imply that long-run growth is \\ governed by population growth
		\item Many other factors have ``level effects'' \\ (e.g., increases in education, R\&D share, misallocation)
		\item But level effects can be large
		\item How much of recent growth is due to such level effects?
		\item What does this suggest about the future of growth? 
	\end{itemize}
\end{frame}


\begin{frame}{Growth Accounting}
	\begin{itemize}
		\item Goods production:
		\[ Y_t = K_t^{\alpha} (Z_t h_t L_{Yt})^{1-\alpha} \] \vspace{-15pt}
		\begin{itemize}
			\item $h_t$ is human capital per person
		\end{itemize}
		\item Productivity:
		\[ Z_t = A_t M_t \] \vspace{-15pt}
		\begin{itemize}
			\item $A_t$ is knowledge
			\item $M_t$ is misallocation
		\end{itemize}
		\item Some manipulation:
		\[y_t = \left(\frac{K_t}{Y_t}\right)^{\alpha/(1-\alpha)}A_t M_t h_t l_t (1-s_t) \] 
	\end{itemize}
\end{frame}


\begin{frame}{Growth Accounting}
	\begin{itemize}
		\itemsep1em 
		\item Ideas Production function:
		\[ \dot{A}(t) = \theta L_A(t)^{\lambda} A(t)^{\phi} \]
		\[ \frac{\dot{A}(t)}{A(t)} = \theta s(t)^{\lambda}L(t)^{\lambda} A(t)^{\phi-1}\]
		\item With constant growth of $A(t)$:
		\[ 0 = \lambda g_s + \lambda g_L - (1-\phi) g_A \]
		\[ g_A = \frac{\lambda}{1-\phi} (g_s + g_L)\]
		\item Jones (2022) assumes $\lambda/(1-\phi) = \lambda/\beta = \gamma =  1/3$ \\ {\small (Results that follow are sensitive to this!)} 
	\end{itemize}
\end{frame}


\begin{frame}{Growth Accounting}
	\begin{figure}
		\centering
		\includegraphics[height = 0.55 \textheight]{FiguresTables/Jones2021Equation15.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Jones (2022).}
\end{frame}


\begin{frame}
	\begin{figure}
		\centering
		\includegraphics[height = 0.8 \textheight]{FiguresTables/Jones2021Figure2.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Jones (2022).}
\end{frame}


\begin{frame}{Growth in Past and Future}
	\begin{itemize}
		\itemsep1em 
		\item In the long run:
		\begin{itemize}
			\item All terms are zero except population growth
			\item 100\% of growth due to population growth
		\end{itemize}
		\item Historically:
		\begin{itemize}
			\item 80\% of growth due to other factors
			\item Only 20\% of growth due to population growth \\ (Sensitive to assumption on $\gamma$.)
		\end{itemize}
	\end{itemize}
\end{frame}


\begin{frame}{Will Growth Slow?}
	\begin{itemize}
		\itemsep1em 
		\item Many sources of growth are temporary:
		\begin{itemize}
			\item Increased education
			\item Higher Emp-Pop ratio
			\item Falling misallocation
			\item Rising research intensity
		\end{itemize}
		\item But some of these might continue for a very long time \\ (e.g., increased research intensity)
		\item Population growth is slowing \\ {\small (Population likely to start shrinking soon!)}
	\end{itemize}
\end{frame}


\begin{frame}
	\begin{figure}
		\centering
		\includegraphics[height = 0.8 \textheight]{FiguresTables/Jones2021Figure4.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Jones (2022).}
\end{frame}


\begin{frame}
	\begin{figure}
		\centering
		\includegraphics[height = 0.8 \textheight]{FiguresTables/Jones2021Figure5.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Jones (2022).}
\end{frame}


\begin{frame}{Might Growth Speed Up?}
	\begin{itemize}
		\itemsep1em 
		\item Finding Einsteins
		\begin{itemize}
			\item Traditionally most people not able to reach their potential \\ as producers of ideas/knowledge
			\item Extreme poverty, cast/class restrictions, discrimination
			\item How many Einsteins and Doudnas have we missed
		\end{itemize}
		\item Automation and Artificial Intelligence
		\begin{itemize}
			\item Interesting discussion in Jones (2022, sec. 6)
			\item Automation of ideas production could even imply a ``singularity'' \\ (explosive growth driven by AGI)
		\end{itemize}
	\end{itemize}
\end{frame}


\end{document}

