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\title[Solow]{Capital Accumulation and Growth: \\ The Solow Model}
\author[Steinsson]{Jón Steinsson}
\institute[]{University of California, Berkeley}
\date[]{Fall 2025}

\begin{document}

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

\begin{frame}{Steady Growth at the Frontier for 150 Years}
	\begin{figure}
		\centering
		\includegraphics[height = 0.8  \textheight]{FiguresTables/Jones2016Figure1.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Jones (2016)}
\end{frame}


\begin{frame}{Uneven Growth Across the World}
	\begin{figure}
		\centering
		\includegraphics[height = 0.8  \textheight]{FiguresTables/MACRO5_FIG03.06.jpg}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Jones (2021)}
\end{frame}


\begin{frame}{Uneven Growth Across the World}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/Jones2016Figure22.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Jones (2016)}
\end{frame}

\begin{frame}{Growth Is a Recent Phenomenon!}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/RealWagesLogScale.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Clark (2010)}
\end{frame}

\begin{frame}{Importance of a Log Scale} \label{logscale}
	\begin{itemize}
		\itemsep1em 
		\item Figures like these often plotted on linear scale \\ to make them more dramatic (hockey stick) \hyperlink{linearscale}{\beamergotobutton{Linear Scale}}
		\item This is misleading. 
		\item Fluctuations before 1800 were large! 
	\end{itemize}
	
	\vspace{30pt}
	{\small (Also Maddison data back thousands of years are ``guestimates'')}
\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?)}
		\item Why did growth begin? 
		\item Why was there no growth before Industrial Revolution?
	\end{itemize}
	
	\vspace{20pt}
	We will focus on first two questions. {\small (210A in the spring covers later two.)}
\end{frame}


\begin{frame}{Mathusian Stagnation / Industrial Revolution}
	\begin{itemize}
		\itemsep1em 
		\item Steinsson,  J. (2021): ``Malthus and Pre-Industrial Stagnation,'' \\ draft textbook chapter. \\ {\small \url{https://eml.berkeley.edu/~jsteinsson/teaching/malthus.pdf} }
		\item Steinsson, J. (2021): ``How Did Growth Begin? The Industrial Revolution and Its Antecedents,'' draft textbook chapter. \\ {\small \url{https://eml.berkeley.edu/~jsteinsson/teaching/originsofgrowth.pdf} } 
	\end{itemize}
\end{frame}

\begin{frame}{Three Textbooks}
	\begin{itemize}
		\itemsep1em 
		\item Romer, D. (2019): \textit{Advanced Macroeconomics}, McGraw Hill, \\ New York, NY. 
		\item Acemoglu, D. (2009): \textit{Introduction to Modern Economic Growth}, Princeton University Press, Princeton, NJ. 
		\item Barro, R.J. and X. Sala-i-Martin (2004): \textit{Economic Growth}, MIT Press, Cambridge, MA. 
	\end{itemize}
\end{frame}

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		\begin{center}
			{\Large \color{white} The Solow Model}
		\end{center}
		
	\end{frame}
}


\begin{frame}{Is Capital Accumulation Key to Growth?}
	\begin{itemize}
		\itemsep1em 
		\item Seems plausible!
		\item Conventional wisdom in 1950s: Yes!
		\item See discussion in Easterly (2002) 
		\item Solow (1956) tackled this question
	\end{itemize}
\end{frame}

\begin{frame}{The Production Function}
	\[ Y(t) = F[K(t), A(t)L(t)] \] \vspace{-15pt}
	\begin{itemize}
		\item $Y(t)$: Output at time $t$
		\item $K(t)$: Capital stock at time $t$
		\item $L(t)$: Labor supply at time $t$
		\item $A(t)$: ``effectiveness of labor'' at time $t$ (aka ``productivity'')
	\end{itemize}
\end{frame}

\begin{frame}{Production Function}
	\[ Y(t) = F[K(t), A(t)L(t)] \] \vspace{-15pt}
	\begin{itemize}
		\item The model is dynamic
		\item Time is continuous
		\item Time only enters production function through inputs
		\item Productivity is ``labor augmenting'' (Harrod neutral)
		\item This last point has traditionally been viewed as important \\ for getting ``balanced growth''
	\end{itemize}
\end{frame}

\begin{frame}{Balanced Growth: Kaldor Facts}
	Kaldor (1963): As per capita income has risen \vspace{5pt}
	\begin{itemize}
		\itemsep0.5em
		\item The capital-output ratio has been roughly constant
		\item Real interest rates have no trend 
		\item The labor and capital share of production have been roughly constant
	\end{itemize}
\end{frame}


\begin{frame}{Roughly Constant Capital-Output Ratio}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/Jones2016Figure3.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Jones (2016)}
\end{frame}


\begin{frame}{Ex Post Real Interest Rate}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/RealInterestRateExPost.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: FRED. 3 month T-bill rate minus 12-month CPI inflation.}
\end{frame}


\begin{frame}{Roughly Constant Labor and Capital Shares}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/Jones2016Figure6.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Jones (2016)}
\end{frame}


\begin{frame}{Forms of Technical Progress}
	\begin{itemize}
		\item Hicks Neutral:
		\[ A(t) F[K(t),L(t)] \]
		{\footnotesize (Ratio of marginal products remains constant for a given $K/L$ ratio)}
		\item Harrod Neutral / Labor-Augmenting:
		\[ F[K(t),A(t)L(t)] \]
		{\footnotesize (Ratio of input shares ($F_{K}K/F_{L}L$) remain constant for a given $K/Y$ ratio)}
		\item Solow Neutral / Capital-Augmenting:
		\[ F[A(t)K(t),L(t)] \]
		{\footnotesize (Ratio of input shares ($F_{K}K/F_{L}L$) remain constant for a given $L/Y$ ratio)}
		\item Some combination of all three also possible
	\end{itemize}
\end{frame}


\begin{frame}{Example: Cobb-Douglas Production Function}
	The Cobb-Douglas production function satisfies all three properties
	
	\vspace{10pt}
	\begin{itemize}
		\itemsep1em 
		\item Hicks Neutral:
		\[ A(t) K(t)^{\alpha} L(t)^{1-\alpha} \]
		\item Harrod Neutral:
		\[  K(t)^{\alpha} [\tilde{A}(t)L(t)]^{1-\alpha} \,\,\, \text{where} \,\,\, \tilde{A}(t) = A(t)^{1/(1-\alpha)} \]
		\item Solow Neutral:
		\[  [\check{A}(t)K(t)]^{\alpha} L(t)^{1-\alpha} \,\,\, \text{where} \,\,\, \check{A}(t) = A(t)^{1/\alpha} \]
	\end{itemize}
\end{frame}


\begin{frame}{Production Functions}
	\begin{itemize}
		\itemsep2em 
		\item Cobb-Douglas:
		\[ Y(t) = A(t) K(t)^{\alpha} L(t)^{1-\alpha} \] \vspace{-15pt}
		\begin{itemize}
			\item $\alpha$ ``weight'' on capital (and capital share)
			\item Elasticity of substitution between capital and labor is 1
		\end{itemize}
		\item Constant elasticity of substitution (CES): 
		\[ Y(t) = \left[ \alpha (A_K(t)K(t))^{\frac{\sigma-1}{\sigma}} + (1-\alpha) (A_L(t)L(t))^{\frac{\sigma-1}{\sigma}} \right]^{\frac{\sigma}{\sigma-1}} \] \vspace{-15pt}
		\begin{itemize}
			\item Elasticity of substitution between capital and labor is $\sigma$
		\end{itemize} 
	\end{itemize}
\end{frame}


\begin{frame}{Uzawa's (1961) Theorem}
	Roughly speaking:
	\begin{itemize}
		\item Balanced growth in the long run is only possible if \\ all technical progress is labor augmenting \\ {\footnotesize (See Acemoglu (2009, sec. 2.7) and Barro-Sala-I-Martin (2004, sec. 1.2.12) for details)}
	\end{itemize}
	
	\vspace{10pt}
	Why balanced growth:
	\begin{itemize}
		\item Empirically: We see a stable $K/Y$ ratio and relatively stable \\ factor shares
		\item Theoretically: Very convenient because model will have a steady state when technical progress is constant
	\end{itemize}
\end{frame}


\begin{frame}{Uzawa's (1961) Theorem}
	Acemoglu (2009, p. 59):
	
	\vspace{10pt}
	\begin{quotation}
		This result is very surprising and troubling, since there are no compelling reasons for why technological progress should take this form. [i.e., be labor augmenting] 
	\end{quotation}
\end{frame}


\begin{frame}{Labor Augmenting or Labor Displacing}
	\begin{itemize}
		\item Textiles: spinning jenny, water frame, mule, mechanized weaving
		\item Power: windmill, water wheel, steam engine, electric motor
		\item Transportation: trains, cars, trucks, airplanes
		\item Agriculture: tractors, combine harvester, fertilizer
		\item Computing: abacus, transistor, microprocessor
	\end{itemize}
	
	\vspace{15pt}
	Why would such technical progress leave labor share unchanged?
\end{frame}


\begin{frame}{Traditional Production Function}
	\[ Y(t) = F[K(t), A(t)L(t)] \] \vspace{-15pt}
	\begin{itemize}
		\item Traditional production function a black box
		\item Lacks descriptive realism -- especially for technical change
		\item Few technologies increase the productivity of a factor in all tasks
		\item More common: Machine takes over one task and makes labor more productive at another task
		\begin{itemize}
			\item Computer: Replaces human computers, makes those using \\ computations more productive
			\item Power loom: Replaces master weaver, creates new tasks for \\ labor to design, build, and operate machines
		\end{itemize}
	\end{itemize}
\end{frame}


\begin{frame}{Task-Based Production Function}
	\begin{itemize}
		\itemsep1em 
		\item Production accomplished by performing a set of tasks \\ {\footnotesize (Zeira 98, Acemoglu-Restrepo 18)}
		\[ Y = \left[ \int_{N-1}^{N} y(z)^{\frac{\sigma-1}{\sigma}} dz \right]^{\frac{\sigma}{\sigma-1}} \]
		\item Each task either technologically automated or not:
		\[ y(z) = \left\{ \begin{array}{ll} A^L \psi^L(z) L(z) + A^k \psi^K(z) K(z) & \text{if} \,\,\, z \in [N-1,I] \\ A^L \psi^L(z) L(z) & \text{if} \,\,\, z \in (I,N]  \end{array}  \right.  \] \vspace{-15pt}
		\begin{itemize}
			\item $\psi^L(z)$ and $\psi^K(z)$ task-specific productivity of factors
		\end{itemize}
		\item Let's order tasks by comparative advantage of labor: $\psi^L(z)/\psi^K(z)$
	\end{itemize}
	{\footnotesize (More general production function for each task possible (realistic).)}
\end{frame}


\begin{frame}{Allocation of Tasks to Factors}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/AcemogluLecture1.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Acemoglu and Restrepo (2019)}
\end{frame}


\begin{frame}{Allocation of Tasks to Factors}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/AcemogluLecture2.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Acemoglu, Economic Growth Lecture 10, 2023 ($\gamma$'s are supposed to be $\psi$'s)}
\end{frame}


\begin{frame}{Allocation of Tasks to Factors}
	\begin{figure}
		\centering
		\includegraphics[height = 0.63  \textheight]{FiguresTables/AcemogluLecture3.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Acemoglu, Economic Growth Lecture 10, 2023 ($\gamma$'s are supposed to be $\psi$'s)}
\end{frame}


\begin{frame}{Allocation of Tasks to Factors}
	\begin{figure}
		\centering
		\includegraphics[height = 0.68  \textheight]{FiguresTables/AcemogluLecture4.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Acemoglu, Economic Growth Lecture 10, 2023 ($\gamma$'s are supposed to be $\psi$'s)}
\end{frame}


\begin{frame}{Allocation of Tasks to Factors}
	\begin{figure}
		\centering
		\includegraphics[height = 0.68  \textheight]{FiguresTables/AcemogluLecture5.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Acemoglu, Economic Growth Lecture 10, 2023 ($\gamma$'s are supposed to be $\psi$'s)}
\end{frame}


\begin{frame}{Allocation of Tasks to Factors}
	\begin{figure}
		\centering
		\includegraphics[height = 0.7  \textheight]{FiguresTables/AcemogluLecture6.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Acemoglu, Economic Growth Lecture 10, 2023 ($\gamma$'s are supposed to be $\psi$'s)}
\end{frame}


\begin{frame}{Derived Production Function}
	\begin{itemize}
		\item Output can be represented by CES production function:
		\[ Y = \Pi(I,N) \left[ \Gamma(I,N) (A_L(t)L(t))^{\frac{\sigma-1}{\sigma}} + (1-\Gamma(I,N)) (A_K(t)K(t))^{\frac{\sigma-1}{\sigma}} \right]^{\frac{\sigma}{\sigma-1}} \]
		\item Labor content of production:
		\[ \Gamma(I,N) = \frac{\int_{I}^{N}\psi^L(z)^{\sigma-1}dz}{\int_{N-I}^{I}\psi^{K}(z)^{\sigma-1}dz+\int_{I}^{N}\psi^L(z)^{\sigma-1}dz} \]
		\item Total factor productivity:
		\[\Pi(I,N) = \left[ \int_{N-1}^{I} \psi^K(z)^{\sigma-1}dz + \int_{I}^{N} \psi^{L}(z)^{\sigma-1} \right]^{\frac{1}{\sigma-1}} \] 
		\item Technical change affects both $\Gamma(I,N)$ and $\Pi(I,N)$ 
	\end{itemize}
\end{frame}


\begin{frame}{Effects of Technical Change on Labor Demand}
	\begin{itemize}
		\itemsep1em 
		\item Displacement effect: Technical change can displace workers 
		\item Productivity effect: Technical change raises productivity which increases demand for remaining workers
		\item Reinstatement effect: Technical change creates new tasks \\ for which labor has a comparative advantage
	\end{itemize}
\end{frame}

\begin{frame}{Automation}
	\begin{itemize}
		\itemsep1em 
		\item Automation displaces workers from certain tasks
		\item Effects on labor demand:
		\begin{itemize}
			\item Displacement effect
			\item Productivity effect
		\end{itemize}
		\item Reduces labor demand (and wages) if displacement effect \\ is stronger than productivity effect
		\item With traditional production function, technical progress \\ increases labor demand
	\end{itemize}
\end{frame}


\begin{frame}{Jones and Liu (2022)}
	\begin{itemize}
		\itemsep1em 
		\item Present a model where all technology is 
		\begin{itemize}
			\item Purely capital-embodied at the micro level
			\item Purely labor-augmenting at the macro level 
		\end{itemize}
		\item Task-based production function (Zeira 98, Acemoglu-Restrepo 18)
		\item Two kinds of innovation:
		\begin{itemize}
			\item More tasks performed by capital (increases capital share)
			\item Innovation on already automated tasks (decreases capital share)
			\begin{itemize}
				\item Innovation reduced price of that task
				\item If tasks are complements, this reduces spending on that task
			\end{itemize}
		\end{itemize}
		\item Combination of the two can yield stable capital share
	\end{itemize}
\end{frame}





\begin{frame}{Returns to Scale}
	\textbf{Definition}: A function $f$ is homogeneous of degree $m$ in $x$ and $y$ if
	\[f(\lambda x, \lambda y, z) = \lambda^m f(x,y,z) \]
	\begin{itemize}
		\item $m < 1$: decreasing returns to scale
		\item $m = 1$: constant returns to scale
		\item $m > 1$: increasing returns to scale
	\end{itemize}
\end{frame}

\begin{frame}{Euler's Theorem}
	\textbf{Euler's Theorem}: If $f$ is homogeneous of degree $m$ in $x$ and $y$:
	\[ m f(x,y,z) = \frac{\partial}{\partial x} f(x,y,z) x + \frac{\partial}{\partial y} f(x,y,z) y \]
	
	\vspace{10pt}
	{\footnotesize (See Acemoglu (2009, p. 29) for a more careful statement of this theorem.) }
\end{frame}


\begin{frame}{Constant Returns to Scale}
	\begin{itemize}
		\item We assume that the production function is constant returns to scale:
		\[ F(cK,cAL) = cF(k,AL) \]
		\item Why? \pause 
		\begin{itemize}
			\itemsep0.5em
			\item Economy large enough that each establishment has reached efficient size (micro returns to scale and gains from specialization exhausted)
			\item Fixed factors (e.g., land) unimportant
			\item Positive and negative externalities between establishments unimportant
			\item $A(t)$ non-rival (can be used many times)
			\item Replication argument: Can build a second identical establishment with double the inputs

		\end{itemize} 
	\end{itemize}
\end{frame}


\begin{frame}{Intensive Form}
	\begin{itemize}
		\item Since
		\[ F(cK,cAL) = cF(K,AL) \]
		\item we can write production function in intensive form:
		\[ \frac{Y}{AL} = \frac{1}{AL} F(K,AL) = F\left( \frac{K}{AL}, 1 \right) \]
		\item Define:
		\begin{itemize}
			\item $k = K/AL$: Capital per effective worker
			\item $y = Y/AL$: Output per effective worker 
		\end{itemize}
		\item Also define: $f(k) = F(k,1)$
		\item Then we have:
		\[ y = f(k) \]
	\end{itemize}
	{\footnotesize (Why do this? ... Will become clear in a few slides.)}
\end{frame}


\begin{frame}{Returns to Capital}
	What do we want to assume about returns to capital? \pause
	
	\vspace{10pt}
	Returns to capital are ...
	\begin{itemize}
		\item Positive: $f'(k) > 0$
		\item Diminishing: $f''(k) < 0$
	\end{itemize}
	
	\vspace{10pt}
	Also ...
	\begin{itemize}
		\item f(0) = 0
		\item Inada conditions:
		\[ \lim_{k \rightarrow 0} f'(k) = \infty \,\,\,\,\, \text{and} \,\,\,\,\, \lim_{k \rightarrow \infty} f'(k) = 0 \] 
	\end{itemize}
\end{frame}


\begin{frame}{Neoclassical Production Function}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/Romer2019Figure11.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Romer (2019)}
\end{frame}


\begin{frame}{Example: Cobb-Douglas }
	\begin{itemize}
		\item If the production function is Cobb-Douglas, we have
		\[y = \frac{Y}{AL} = \frac{1}{AL} K^{\alpha} (AL)^{1-\alpha} = \left( \frac{K}{AL} \right)^{\alpha} = k^{\alpha}. \]
		So, we have: 
		\[ y = k^{\alpha} \]
		\item This function satisfies all the conditions we have specified \\ on previous slides
	\end{itemize}
\end{frame}

\begin{frame}{Cambridge Capital Controversy}
	\begin{itemize}
		\itemsep1em 
		\item Early post-WWII debate between (mostly) British and \\ (mostly) US economists
		\item Does it make sense to talk about aggregate capital?
		\item Do lower interest rates lead to higher capital/labor ratios?
		\item Outcome:
		\begin{itemize}
			\item Various pathologies possible
			\item Similar to Giffen goods in consumption theory
			\item Not clear any of this is practically important
		\end{itemize}
	\end{itemize}
\end{frame}


\begin{frame}{Cambridge Capital Controversy}
	Cambridge, U.K.:
	\begin{itemize}
		\item Harcourt, G.C. (1969): ``Some Cambridge Controversies in the Theory of Capital,'' \textit{Journal of Economic Literature}, 7(2), 369-405.
		\item Cohen, A.J. and G.C. Harcourt (2003): ``Whatever Happened to the Cambridge Capital Theory Controversies?'' \textit{Journal of Economic Perspectives}, 17(1), 199-214.
	\end{itemize}
	Cambridge, U.S.:
	\begin{itemize}
		\item Samuelson, P.A. (1966): ``A Summing Up,'' \textit{Quarterly Journal of Economics}, 80(4), 568-583.
		\item Stiglitz, J.E. (1974): ``The Cambridge-Cambridge Controversy in the Theory of Capital: A View from New Haven,'' \textit{Journal of Political Economy}, 82(4), 893-903.
	\end{itemize}
\end{frame}

 

\begin{frame}{What Happens to the Output?}
	\begin{itemize}
		\itemsep1em 
		\item Output is divided between consumption and investment:
		\[ Y(t) = C(t) + I(t) \]
		\item How much is invested?
		\item Simplifying assumption: Constant savings rate
		\[ I(t) = s Y(t)\] 
	\end{itemize}
	{ \small (We will introduce optimizing households in Ramsey model) }
\end{frame}

\begin{frame}{Evolution of Capital}
	\begin{align*}
		\dot K(t) & = I(t) - \delta K(t) \\
		          & = s Y(t) - \delta K(t)
	\end{align*} %\vspace{-10pt}  
	\hspace{17pt} ($\dot K(t) = d K(t) / dt$)
	\begin{itemize}
		\itemsep1em 
		\item Each instant:
		\begin{itemize}
			\item New investment adds to capital stock
			\item Existing capital depreciates by some fraction (per unit time) 
		\end{itemize}
		\item Change in capital stock is the difference between these two
	\end{itemize}
\end{frame}

\begin{frame}{Labor and Productivity Exogenous}
	\begin{itemize}
		\item Labor and productivity grow at constant rates:
		\[ \dot L(t) = n L(t) \]
		\[ \dot A(t) = g A(t)\]
		\item Notice that
		\[ \frac{d \log X(t)}{dt} = \frac{d \log X(t)}{d X(t)} \frac{d X(t)}{dt} = \frac{\dot X(t)}{X(t)} \]
		where $\log$ denotes the natural log
		\[ \frac{d \log L(t)}{dt} = \frac{\dot L(t)}{L(t)} = n \] 
		\[\log L(t) = \log L(0) + nt \]
		\[ L(t) = L(0)e^{nt}\]
		and similarly for $A(t)$. 
	\end{itemize}
\end{frame}


\begin{frame}{Full Solow Model}
	\[Y(t) = F[K(t),A(t)L(t)] \]
	\[ Y(t) = C(t) + I(t) \]
	\[ I(t) = s Y(t)\]
	\[ \dot K(t) = I(t) - \delta K(t) \]
	\[ \dot L(t) = n L(t) \]
	\[ \dot A(t) = g A(t)\] \vspace{-10pt}
	\begin{itemize}
		\item Initial Conditions: $K(0)$, $A(0)$, $L(0)$ given
		\item Goal: Solve for evolution of $K(t)$, $Y(t)$, $C(t)$, $I(t)$, $L(t)$, $A(t)$ 
	\end{itemize}
\end{frame}


\begin{frame}{About Models}
	\begin{itemize}
		\itemsep1em 
		\item Solow model is a gross simplification
		\item Not necessarily a defect
		\item Real world is fully realistic, but too complicated to understand
		\item Simple models can provide insight about specific issues
		\item But may cause ``theory-induced blindness''
		\item Kahneman: ``Once you have accepted a theory, it is extraordinarily difficult to notice its flaws.''
		\item Fully realistic model not insightful but would allow for calculation of counterfactuals and the analysis of policy experiments
	\end{itemize}
\end{frame}


\begin{frame}{About Models}
	Two uses of models: \vspace{5pt}
	\begin{itemize}
		\itemsep1em 
		\item Provide insight about mechanisms
		\begin{itemize}
			\item Such models must be (relatively) simple
			\item Unlikely to be good guides to real-world counterfactuals
		\end{itemize}
		\item Provide a basis for policy evaluation 
		\begin{itemize}
			\item Such models need not be insightful
			\item But they must be ``realistic''
		\end{itemize}
	\end{itemize}
	
	\vspace{10pt}
	Important to keep this distinction clear
\end{frame}


\begin{frame}{Finding a Steady State}
	\begin{itemize}
		\itemsep1em 
		\item When solving a dynamic system of equations, often useful to find \\ a steady state
		\item A \textbf{stable} steady state is a point the system stays at if unperturbed \\ and returns to if perturbed by a small amount
		\item Since $L(t)$ and $A(t)$ are growing, no steady state in the original variables
		\item Key to finding a steady state to work with transformed variables:
		\[ y(t) = \frac{Y(t)}{A(t)L(t)} \hspace{40pt} k(t) = \frac{K(t)}{A(t)L(t)} \] 
	\end{itemize}
\end{frame}


\begin{frame}{Dynamics of $k(t)$}
	\begin{itemize}
		\item Using the chain rule we have that
		\begin{align*}
			\dot k(t) & = \frac{\dot K(t)}{A(t)L(t)} - \frac{K(t)}{[A(t)L(t)]^2}[\dot A(t)L(t) + A(t)\dot L(t)] \\
			          & = \frac{\dot K(t)}{A(t)L(t)} - \frac{K(t)}{A(t)L(t)} \frac{\dot L(t)}{L(t)} - \frac{K(t)}{A(t)L(t)} \frac{\dot A(t)}{A(t)}
		\end{align*}
		\item Using $\dot L/L = n$, $\dot A/A = g$, and $\dot K = sY - \delta K$ we have that
		\[ \dot k(t) = \frac{sY(t) - \delta K(t)}{A(t)L(t)} - n k(t) - g k(t)  \] 
		\item Using $y = f(k)$ we have that
		\[ \dot k(t) = sf(k(t)) - (n+g+\delta)k(t)  \]
	\end{itemize}
\end{frame}


\begin{frame}{Dynamics of $k(t)$}
	\[ \dot k(t) = sf(k(t)) - (n+g+\delta)k(t)  \] \vspace{-10pt}
	\begin{itemize}
		\itemsep1em 
		\item Rate of change of $k(t)$ difference between:
		\begin{itemize}
			\item Actual investment: $sf(k(t))$
			\item Break-even investment: $(n+g+\delta)k(t)$
		\end{itemize}
		\item Notice that break-even investment determined by:
		\begin{itemize}
			\item Population growth: $n$
			\item Productivity growth: $g$
			\item Depreciation: $\delta$
		\end{itemize}
		\item Intuition: capital per effective worker must keep up with amount of effective labor (which is growing due to $n$ and $g$)
	\end{itemize}
\end{frame}

\begin{frame}{Actual and Break-Even Investment}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/Romer2019Figure12.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Romer (2019)}
\end{frame}


\begin{frame}{Phase Diagram for $k(t)$}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/Romer2019Figure13.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Romer (2019)}
\end{frame}


\begin{frame}{Economy Converges to a Steady State $k^{*}$}
	\begin{itemize}
		\itemsep1em 
		\item Inada conditions and $f''(k) < 0$ imply that actual investment and break-even investment lines cross once \\ {\small (with actual investment crossing from above)}
		\item This point is denoted $k^{*}$
		\item $k^{*}$ is a steady state for $k(t)$
		\item Economy converges to $k^{*}$ globally \\ {\small (i.e., from any (positive) starting point)}
	\end{itemize}
\end{frame}


\begin{frame}{Balanced Growth Path}
	\begin{itemize}
		\itemsep1em 
		\item At steady state $k(t)$ is constant
		\item This implies that $K = ALk$ grows at a rate $n+g$
		\item Since both $K$ and $AL$ grow at $n+g$, $Y$ also grows at rate $n+g$
		\item Furthermore, $K/L$ and $Y/L$ grow at rate $g$
		\item Economy converges to a balanced growth path
	\end{itemize}
	
	\vspace{20pt}
	{\small These conclusion flow from fact that the growth rate of the product of two variables \\ is the sum of their growth rates. See, Problem 1.1 in Romer (2019).}
\end{frame}


\begin{frame}{First Lesson from Solow Model}
	\begin{itemize}
		\itemsep1em 
		\item Capital accumulation cannot serve as a source of \\ long-run growth in living standards
		\begin{itemize}
			\item If $g = 0$, growth in $Y/L$ is zero
		\end{itemize}
		\item Why? \pause Because of diminishing returns to capital.
		\begin{itemize}
			\item Diminishing returns mean actual investment eventually cannot keep up with break-even investment 
			\item This gives rise to a steady state with property listed above
		\end{itemize}
		\item Long-run growth must come from $A(t)$
	\end{itemize}
\end{frame}


\begin{frame}{Effects of Changes in Fundamental Parameters}
	\begin{itemize}
		\itemsep1em 
		\item One can use the Solow model to think about changes in:
		\begin{itemize}
			\item The savings rate $s$
			\item The population growth rate $n$
			\item The growth rate of technology $g$
			\item The depreciation rate $\delta$
		\end{itemize}
		\item Such exercises are ``other things equal'' type exercises
		\item Let's consider a permanent increase in the savings rate
		\item How does this affect actual and break-even investment curves?
	\end{itemize}
\end{frame}


\begin{frame}{Increase in the Savings Rate}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/Romer2019Figure14.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Romer (2019)}
\end{frame}


\begin{frame}{Increase in the Savings Rate}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/Romer2019Figure15_1.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Romer (2019)}
\end{frame}


\begin{frame}{Increase in the Savings Rate}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/Romer2019Figure15_2.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Romer (2019)}
\end{frame}


\begin{frame}{Increase in Savings Rate}
	\begin{itemize}
		\itemsep1em 
		\item Increase in savings rate has a ``level effect'' on per capita output
		\item It does not have a ``growth effect''
	\end{itemize}
\end{frame}


\begin{frame}{Destruction of Capital}
	\begin{itemize}
		\itemsep1em 
		\item Suppose half the capital stock of a country is destroyed
		\item What does Solow model predict about output
		\begin{itemize}
			\item In the short run?
			\item In the long run?
		\end{itemize} \pause
		\item When has this happened in the real world?
	\end{itemize}
\end{frame}


\begin{frame}
	\begin{figure}
		\centering
		\includegraphics[height = 0.85  \textheight]{FiguresTables/DestructionOfCapitalGermany1.pdf}
	\end{figure}
\end{frame}


\begin{frame}
	\begin{figure}
		\centering
		\includegraphics[height = 0.85  \textheight]{FiguresTables/DestructionOfCapitalGermany2.pdf}
	\end{figure}
\end{frame}


\begin{frame}
	\begin{figure}
		\centering
		\includegraphics[height = 0.85  \textheight]{FiguresTables/DestructionOfCapitalJapan.pdf}
	\end{figure}
\end{frame}


\begin{frame}
	\begin{figure}
		\centering
		\includegraphics[height = 0.85  \textheight]{FiguresTables/DestructionOfCapitalUS.pdf}
	\end{figure}
\end{frame}


{  \setbeamercolor{background canvas}{bg=lightblue}
	\begin{frame}
		\addtocounter{framenumber}{-1}
		\thispagestyle{empty}
		
		\begin{center}
			{\Large \color{white} Transition Dynamics}
		\end{center}
		
	\end{frame}
}


\begin{frame}{Transition Dynamics in the Solow Model}
	\begin{itemize}
		\itemsep1em 
		\item Our focus has been on long run effects
		\item Solow model also has interesting implications about ``short run''
	\end{itemize}
\end{frame}



\begin{frame}{Transition Dynamics in the Solow Model}
	\begin{itemize}
		\item Start with 
		\[ \dot k(t) = sf(k(t)) - (n+g+\delta)k(t)  \]
		\item Divide by $k(t)$:
		\[ \frac{\dot k(t)}{k(t)} = \frac{sf(k(t))}{k(t)} - (n+g+\delta)  \]
		\item Left-hand-side is growth rate of capital
		\item $(n+g+\delta)$ is constant as a function of $k(t)$
		\item While 
		\[ \lim_{k \rightarrow 0} \frac{sf(k(t))}{k(t)} = \infty \;\;\;\;\;\;\;\; \lim_{k \rightarrow \infty} \frac{sf(k(t))}{k(t)} = 0 \] 
		\[ \frac{d}{dk} \frac{sf(k)}{k} = - s\frac{f(k)/k - f'(k)}{k} < 0 \]
		{\footnotesize (numerator is average product of capital minus marginal product of capital)}
	\end{itemize}
\end{frame}


\begin{frame}{Transition Dynamics}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/BarroSalaIMartin2004Figure14.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Barro and Sala-I-Martin (2004). Figure is for $g = 0$. Adding $g>0$ would just shift up horizontal line.}
\end{frame}

\begin{frame}{Transitional Growth Rates}
	\begin{itemize}
		\item Differentiate $y(t) = f(k(t))$ with respect to $t$
		\[\dot y(t) = f'(k(t)) \dot k(t) \]
		\item Divide through by $y(t)$:
		\[ \frac{\dot y(t)}{y(t)} = \frac{f'(k(t))k(t)}{f(k(t))} \frac{\dot k(t)}{k(t)} \]  
		\item Let $g_x$ denote the growth rate of $x_t$ and $\alpha_K(k(t)) =  f'(k(t))k(t)/f(k(t))$
		\[ g_y = \alpha_K(k(t)) g_k \] 
		{\small ($\alpha_K(k(t))$ is the elasticity of output with respect to capital.)}
		\item Growth rate of output is proportional to growth rate of capital
	\end{itemize}
\end{frame}


\begin{frame}{Second Lesson from Solow Model}
	\begin{itemize}
		\itemsep1em 
		\item Countries that are below \textbf{their} steady state level of capital/output should grow faster than countries that are above \textbf{their} steady state. 
		\item If countries share same fundamentals, Solow model predicts \\ \textit{absolute convergence}
		\item More generally, Solow model predicts \textit{conditional convergence}
	\end{itemize}
\end{frame}


\begin{frame}{Baumol (1986)}
	\begin{itemize}
		\itemsep1em 
		\item Analyzed data for 16 industrialized countries for which \\ long historical data were available
		\item Estimated:
		\[ \log \tilde{y}_{i,1979} - \log \tilde{y}_{i,1870} = a + b \log \tilde{y}_{i,1870} + \epsilon_i \]
		where $\tilde{y}_{i,t}$ denotes output per person in country $i$ at time $t$ 
		\item Negative $b$ indicates convergence (initial poor grow faster)
	\end{itemize}
\end{frame}


\begin{frame}{Baumol (1986)}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/Romer2019Figure17.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Romer (2019)}
\end{frame}


\begin{frame}{De Long (1988)}
	\begin{itemize}
		\itemsep1em 
		\item De Long (1988) presented two important critiques of Baumol (1986)
		\item Sample selection:
		\begin{itemize}
			\item Baumol chose countries that were ex post rich
			\item Any difference in initial conditions will yield convergence
			\item Data more likely to be available for ex post successful countries
			\item De Long selects countries based on initial GDP per capita
		\end{itemize}
		\item Measurement error:
		\begin{itemize}
			\item Initial income shows up both on LHS and RHS
			\item Measurement error in initial income creates bias toward convergence 
		\end{itemize} 
	\end{itemize}
\end{frame}


\begin{frame}{De Long (1988)}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/Romer2019Figure18.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Romer (2019)}
\end{frame}


\begin{frame}{OECD Post-1960}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/Jones2016Figure25.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Jones (2016)}
\end{frame}


\begin{frame}{U.S. States Post-1880}
	\begin{figure}
		\centering
		\includegraphics[height = 0.8  \textheight]{FiguresTables/BarroSalaIMartin1991Figure1.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Barro and Sala-I-Martin (1991)}
\end{frame}


\begin{frame}{All Countries Post-1960}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/Jones2016Figure26.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Jones (2016)}
\end{frame}


\begin{frame}{Conditional Convergence}
	\begin{itemize}
		\item Solow model implies:
		\[ \dot k(t) = sf(k(t)) - (n+g+\delta)k(t)  \]
		\item If $f(k(t)) = k(t)^{\alpha}$, steady state:
		\[ k^{*} =  \left(\frac{s}{n+g+\delta} \right)^{1/(1-\alpha)}\] 
	\end{itemize}
\end{frame}


\begin{frame}{Conditional Convergence}
	\begin{itemize}
		\itemsep1em 
		\item But $k = K/AL$ is not observable ($A$ is not observable)
		\item Let's rewrite the steady state in terms of $K/L$
		\[ \left(\frac{K}{L}\right)^{*} =  A \left(\frac{s}{n+g+\delta} \right)^{1/(1-\alpha)}\]
		\item Model implies convergence conditional on: $A$, $s$, $n$, $g$, $\delta$
	\end{itemize}
\end{frame}


\begin{frame}{Mankiw, Romer, and Weil (1992)}
	\begin{figure}
		\centering
		\includegraphics[height = 0.7  \textheight]{FiguresTables/MankiwRomerWeil1992Table3.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Mankiw, Romer, Weil (1992)}
\end{frame}


\begin{frame}{Mankiw, Romer, and Weil (1992)}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/MankiwRomerWeil1992Table4.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Mankiw, Romer, Weil (1992)}
\end{frame}


\begin{frame}{Mankiw, Romer, and Weil (1992)}
	\begin{figure}
		\centering
		\includegraphics[height = 0.8  \textheight]{FiguresTables/MankiwRomerWeil1992Table5.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Mankiw, Romer, Weil (1992)}
\end{frame}


\begin{frame}{Mankiw, Romer, and Weil (1992)}
	\begin{figure}
		\centering
		\includegraphics[height = 0.65  \textheight]{FiguresTables/MankiwRomerWeil1992Figure1A.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Mankiw, Romer, Weil (1992)}
\end{frame}


\begin{frame}{Mankiw, Romer, and Weil (1992)}
	\begin{figure}
		\centering
		\includegraphics[height = 0.65  \textheight]{FiguresTables/MankiwRomerWeil1992Figure1C.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Mankiw, Romer, Weil (1992)}
\end{frame}


\begin{frame}{Determinants of Growth}
	\[ \left(\frac{K}{L}\right)^{*} =  A \left(\frac{s}{n+g+\delta} \right)^{1/(1-\alpha)}\] \vspace{-10pt}
	\begin{itemize}
		\itemsep1em 
		\item Mankiw-Romer-Weil 92 condition on $s$, $n$, schooling
		\item But what about $A$?
		\item Perhaps differences in $A$ are not needed to explain \\ cross-country growth
		\item We will come back to this when we consider development accounting \\ in a few lectures
	\end{itemize}
\end{frame}


\begin{frame}{Convergence}
	\begin{itemize}
		\itemsep1em 
		\item Unconditional convergence:
		\begin{itemize}
			\item Within OECD countries
			\item Within US states, Japanese prefectures, etc.
		\end{itemize}
		\item Conditional convergence across all countries
		\item Is convergence the dominant fact about growth?
	\end{itemize}
\end{frame}


\begin{frame}{Great Divergence}
	\begin{itemize}
		\itemsep1em 
		\item Zooming out in time there has clearly been huge divergence
		\item Before 1500, most countries relatively equally poor
		\item Then some countries became rich and others didn't
		\item Pritchett (1997): Divergence, Big Time
	\end{itemize}
\end{frame}


\begin{frame}{First Great Divergence}
	\begin{figure}
		\centering
		\includegraphics[height = 0.6  \textheight]{FiguresTables/AcemogluJohnsonRobinson2005Figure1A.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Acemoglu, Johnson, and Robinson (2005)}
\end{frame}


\begin{frame}{Pritchett (1997)}
	\begin{itemize}
		\itemsep1em 
		\item Lack of reliable data for less developed countries in 19th century \\ is a problem for divergence calculations 
		\item But we can put a conservative lower bound on per capita GDP
		\item Argues that \$250 PPP is conservative (1985 dollars)
		\begin{itemize}
			\item Lower than lowest sustained measurements on record
			\item Less than enough to buy 2000 calories a day
		\end{itemize}
		\item Backcasts current GDP per capita subject to lower bound
	\end{itemize}
\end{frame}


\begin{frame}{Divergence, Big Time}
	\begin{figure}
		\centering
		\includegraphics[height = 0.8  \textheight]{FiguresTables/Pritchett1997Figure1.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Pritchett (1997)}
\end{frame}


\begin{frame}{Divergence, Big Time}
	\begin{figure}
		\centering
		\includegraphics[height = 0.76  \textheight]{FiguresTables/Pritchett1997Table2.pdf}
	\end{figure}
	\vspace{-4.5mm}
	{\scriptsize Source: Pritchett (1997). 1870 estimates for LDC calculated by ``smushing'' distribution between \\ \vspace{-7pt} lower bound and US.}
\end{frame}


\begin{frame}{Sala-i-Martin (2006)}
	\begin{itemize}
		\itemsep1em 
		\item Most work on convergence focuses on countries
		\item But for welfare calculations we should focus on people
		\item Two complications:
		\begin{itemize}
			\item Countries are of vastly different sizes \\ (e.g., China more populous than all of Africa ($\approx$50 countries))
			\item There is a distribution of income within countries
		\end{itemize}
		\item Attempts to calculate World Distribution of Income from 1970-2000
		\begin{itemize}
			\item Mean income level from NIPA data for each country
			\item Uses micro-surveys to construct distribution within country
		\end{itemize}
		\item Subtitle of paper: ``Falling Poverty, and ... Convergence, Period''
	\end{itemize}
\end{frame}


\begin{frame}{Divergence at Country Level}
	\begin{figure}
		\centering
		\includegraphics[height = 0.8  \textheight]{FiguresTables/Salaimartin2006Figure1a.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Sala-i-Martin (2006). Growth 1970-2000 on level in 1970.}
\end{frame}


\begin{frame}{Convergence at Person Level}
	\begin{figure}
		\centering
		\includegraphics[height = 0.8  \textheight]{FiguresTables/Salaimartin2006Figure1b.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Sala-i-Martin (2006). Growth 1970-2000 on level in 1970.}
\end{frame}


\begin{frame}{Distribution of Income in China}
	\begin{figure}
		\centering
		\includegraphics[height = 0.7  \textheight]{FiguresTables/Salaimartin2006Figure2a.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Sala-i-Martin (2006)}
\end{frame}


\begin{frame}{Distribution of Income in India}
	\begin{figure}
		\centering
		\includegraphics[height = 0.7  \textheight]{FiguresTables/Salaimartin2006Figure2b.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Sala-i-Martin (2006)}
\end{frame}


\begin{frame}{Distribution of Income in Brazil}
	\begin{figure}
		\centering
		\includegraphics[height = 0.7  \textheight]{FiguresTables/Salaimartin2006Figure2e.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Sala-i-Martin (2006)}
\end{frame}


\begin{frame}{World Distribution of Income in 1970}
	\begin{figure}
		\centering
		\includegraphics[height = 0.7  \textheight]{FiguresTables/Salaimartin2006Figure3a.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Sala-i-Martin (2006)}
\end{frame}


\begin{frame}{World Distribution of Income in 2000}
	\begin{figure}
		\centering
		\includegraphics[height = 0.7  \textheight]{FiguresTables/Salaimartin2006Figure3b.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Sala-i-Martin (2006)}
\end{frame}


\begin{frame}{Falling Poverty}
	\begin{figure}
		\centering
		\includegraphics[height = 0.7  \textheight]{FiguresTables/Salaimartin2006Figure6.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Sala-i-Martin (2006)}
\end{frame}


\begin{frame}{Falling Poverty}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/Salaimartin2006Table1.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Sala-i-Martin (2006)}
\end{frame}


\begin{frame}{Falling Poverty (except in Africa)}
	\begin{figure}
		\centering
		\includegraphics[height = 0.65  \textheight]{FiguresTables/Salaimartin2006Table2.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Sala-i-Martin (2006)}
\end{frame}


\begin{frame}{Falling Poverty (except in Africa)}
	\begin{figure}
		\centering
		\includegraphics[height = 0.7  \textheight]{FiguresTables/Salaimartin2006Figure7.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Sala-i-Martin (2006)}
\end{frame}


\begin{frame}{High Growth in Africa since 2000}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/ACO2021Figure1.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Archibong, Coulibaly, Okonjo-Iweala (2021)}
\end{frame}


\begin{frame}{Convergence, Period}
	\begin{figure}
		\centering
		\includegraphics[height = 0.7  \textheight]{FiguresTables/Salaimartin2006Figure8.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Sala-i-Martin (2006)}
\end{frame}


\begin{frame}{World Inequality Database}
	\begin{itemize}
		\itemsep1em 
		\item New wave of research on income inequality since 2000
		\item Combines data from: national accounts, tax data, household surveys, inheritance records, etc. 
		\item Tax data crucial to capture income shares at the top of the distribution
		\item Key researchers include: Piketty, Saez, and Zucman
		\item Have developed World Inequality Database
	\end{itemize}
\end{frame}


\begin{frame}{Global Income Inequality}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/ChancelPiketty2021Figure1.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Chancel and Piketty (2021)}
\end{frame}


\begin{frame}{Global Income Inequality}
	\begin{figure}
		\centering
		\includegraphics[height = 0.8  \textheight]{FiguresTables/ChancelPiketty2021Figure2.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Chancel and Piketty (2021)}
\end{frame}


\begin{frame}{Global Income Inequality}
	\begin{figure}
		\centering
		\includegraphics[height = 0.8  \textheight]{FiguresTables/ChancelPiketty2021Figure3.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Chancel and Piketty (2021)}
\end{frame}


\begin{frame}{Global Income Inequality}
	\begin{figure}
		\centering
		\includegraphics[height = 0.8  \textheight]{FiguresTables/ChancelPiketty2021Figure4.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Chancel and Piketty (2021)}
\end{frame}


\begin{frame}{Household Survey Underreporting}
	\begin{itemize}
		\item World Inequality Database adds tax data
		\begin{itemize}
			\item Tax data great for rich countries
			\item Essentially no tax data for Africa
			\item 0.5\% of tax units in China, 2\% of tax units in India
		\end{itemize}
		\item Household surveys suffer from underreporting
		\begin{itemize}
			\item Earlier work adjusts mean ... but may affect distribution
			\item Pinkovskiy et al. (2024) use regional data to assess effect on distribution
			\item Growing importance of underreporting of the poor
		\end{itemize}
	\end{itemize}
\end{frame}


\begin{frame}{Global Gini Coefficient}
	\begin{figure}
		\centering
		\includegraphics[height = 0.8  \textheight]{FiguresTables/Pinkovskiyetal2024Figure8.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Pinkovskiy et al. (2024). Blue lines are their preferred estimates. Green lines are WID.}
\end{frame}


\begin{frame}{Global Poverty Rate}
	\begin{figure}
		\centering
		\includegraphics[height = 0.8  \textheight]{FiguresTables/Pinkovskiyetal2024Figure9.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Pinkovskiy et al. (2024). Blue lines are their preferred estimates. Green lines are WID.}
\end{frame}


\begin{frame}{Speed of Convergence}
	\begin{itemize}
		\itemsep1em 
		\item What does Solow model imply about speed of convergence?
		\item If speed of convergence is fast:
		\begin{itemize}
			\item Most countries will be close to steady state \\ (already mostly converged)
			\item We can focus on steady state analysis
		\end{itemize}
		\item Also interesting as a possible test of the model
	\end{itemize}
\end{frame}


\begin{frame}{Speed of Convergence}
	\begin{itemize}
		\itemsep1em 
		\item Start with:
		\[ \dot k(t) = sf(k(t)) - (n+g+\delta)k(t)  \]
		\item So $\dot k(t)$ is a function of $k(t)$
		\item Let's write this as $\dot k(k)$ {\small (dropping dependence on $t$ for notational simplicity)}
		\item A first-order Taylor series approximation of $\dot k(k)$ around $k^{*}$ is:
		\[ \dot k \simeq \left[ \left. \frac{\partial \dot k(k)}{\partial k} \right\vert_{k=k^{*}}  \right](k-k^{*}) \]
		{\scriptsize ($\dot k$ is zero at $k^*$)} \vspace{-10pt}
		\item Let's denote $\lambda = - \partial \dot k(k)/\partial k \vert_{k=k^{*}}$ which means we have
		\[ \dot k(t) \simeq -\lambda (k(t)-k^{*}) \] 
	\end{itemize}
\end{frame}


\begin{frame}{Speed of Convergence}
	\begin{itemize}
		\item Linear first-order differential equation:
		\[ \dot k(t) \simeq -\lambda (k(t)-k^{*}) \]
		\item Solution:
		\[ k(t) - k^{*} \simeq e^{-\lambda t} [k(0) - k^{*}] \]
		\item So, $\lambda$ is rate of convergence
		\item Half-life:
		\[ 0.5 = e^{-\lambda t} \]
		\[ t = -\log(0.5)/\lambda \simeq 0.69/\lambda\]   
	\end{itemize}
\end{frame}


\begin{frame}{Speed of Convergence}
	\begin{itemize}
		\itemsep1em 
		\item Using:
		\[ \dot k(k) = sf(k) - (n+g+\delta)k  \]
		we get that 
		\begin{align*}
			\lambda = -\left[ \left. \frac{\partial \dot k(k)}{\partial k} \right\vert_{k=k^{*}}  \right] & = -[sf'(k^{*}) - (n+g+\delta)] \\
			& = (n+g+\delta) - \frac{(n+g+\delta)k^{*}f'(k^{*})}{f(k^{*})} \\
			& = [1-\alpha_{K}(k^{*})](n+g+\delta)
		\end{align*}
		\item Speed of convergence of output is the same as capital
	\end{itemize}
\end{frame}


\begin{frame}{Speed of Convergence}
	\begin{itemize}
		\item Solow model implies that speed of convergence is
		\[ \lambda = [1-\alpha_{K}(k^{*})](n+g+\delta) \]
		Rough calibration:
		\begin{itemize}
			\item Technological growth: $g = 0.02$
			\item Population growth: $n = 0.01$
			\item Depreciation: $\delta = 0.05$
			\item Capital share: $\alpha_{K}(k^{*}) = 1/3$
		\end{itemize}
		\[ \lambda = \frac{2}{3} (0.01 + 0.02 + 0.05 ) = 0.053 \]
		\item This implies a half-life of 13 years
		\item Very fast convergence!! 
	\end{itemize}
\end{frame}

\begin{frame}{Speed of Convergence in the Data}
	\begin{itemize}
		\itemsep1em 
		\item To measure speed of convergence in the data, must run convergence regressions in terms of annual growth rates
		\item Barro and Sala-I-Martin (1991,1992) consider:
		\[ \frac{1}{T} \log \left( \frac{y_{i,t}}{y_{i,t-T}} \right) = a - (1-e^{-\beta T})\frac{1}{T}\log y_{i,t-T} + \text{other variables}  \]
		\item In this case, $\beta$ is the annual rate of convergence
	\end{itemize}
\end{frame}


\begin{frame}
	\begin{figure}
		\centering
		\includegraphics[height = 0.9 \textheight]{FiguresTables/BarroSalaIMartin1991Table1.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Barro and Sala-I-Martin (1991)}
\end{frame}


\begin{frame}{Speed of Convergence}
	\begin{figure}
		\centering
		\includegraphics[height = 0.7 \textheight]{FiguresTables/BarroSalaIMartin1992Table3.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Barro and Sala-I-Martin (1992)}
\end{frame}


\begin{frame}{Speed of Convergence in the Data}
	\begin{itemize}
		\itemsep1em 
		\item Barro's ``iron law of convergence'': 2\% per year
		\item This implies a half-life of 35 years
		\item Takes 115 years for 90\% of convergence to occur 
		\item Convergence is very slow in practice!!
	\end{itemize}
\end{frame}

\begin{frame}{Reconciling Model and Data}
	\begin{itemize}
		\itemsep1em 
		\item Convergence in basic Solow model way too fast:
		\[ \lambda = [1-\alpha_{K}(k^{*})](n+g+\delta) \]
		\item One way to reconcile model and data is to raise the value of $\alpha_{K}(k^{*})$
		\item if $\alpha_{K}(k^{*}) \simeq 0.75$ then convergence will be close to 2\% per year
		\item $\alpha_{K}(k^{*})$ is the capital share (if markets are competitive)
		\item High $\alpha_{K}(k^{*})$ may make sense if one includes human capital
	\end{itemize}
\end{frame}

\begin{frame}{Kremer, Willis, You (2021)}
	\begin{itemize}
		\itemsep1em 
		\item Revisit convergence after 25 years
		\item Absolute convergence since 2000
		\item Why? Proximate answer: Fundamentals have converged \\ (i.e., $A$, $s$, $n$, etc.)
		\item Leaves deeper question of why fundamentals have converged
	\end{itemize}
\end{frame}


\begin{frame}{Converging to Convergence}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/KremerWillisYou2021Figure1.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Kremer, Willis, You (2021)}
\end{frame}


\begin{frame}{Converging to Convergence}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/KremerWillisYou2021Figure2a.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Kremer, Willis, You (2021)}
\end{frame}


\begin{frame}{Converging to Convergence}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/KremerWillisYou2021Figure3.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Kremer, Willis, You (2021)}
\end{frame}


\begin{frame}{Converging to Convergence}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/KremerWillisYou2021Figure4.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Kremer, Willis, You (2021)}
\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% APPENDIX
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\appendix

{  \setbeamercolor{background canvas}{bg=lightblue}
	\begin{frame}
		\addtocounter{framenumber}{-1}
		\thispagestyle{empty}
		
		\begin{center}
			{\Large \color{white} Appendix}
		\end{center}
		
	\end{frame}
}

\begin{frame}{Beware the Linear Scale!} \label{linearscale}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/RealWages.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Clark (2010)}
	\hyperlink{logscale}{\beamerreturnbutton{Back}}
\end{frame}

\end{document}

