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

\begin{document}

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

\begin{frame}{Growth Accounting: Basics}
	\[ Y = F(K,AL) \] \vspace{-10pt} 
	\begin{itemize}
		\itemsep1em 
		\item How much of growth is ``due to'':
		\begin{itemize}
			\item Growth in inputs (capital, labor, etc.)
			\item Growth in technology ($A$)
		\end{itemize}
		\item First step in understanding determinants of growth since \\ it does not attempt to explain growth in inputs
		\item Exercise that goes back to Abramovitz (1956) and Solow (1957)
	\end{itemize}
\end{frame}

\begin{frame}{Growth Accounting: Basics}
	\begin{itemize}
		\item Starting point:
		\[ Y(t) = F[K(t),A(t)L(t)] \]
		\item Differentiate with respect to time
		\[ \dot{Y}(t) = \frac{\partial Y(t)}{\partial K(t)} \dot{K}(t) + \frac{\partial Y(t)}{\partial L(t)} \dot{L}(t) + \frac{\partial Y(t)}{\partial A(t)} \dot{A}(t) \]
		where $\frac{\partial Y}{\partial L}$ denotes $\frac{\partial Y}{\partial AL}A$ and $\frac{\partial Y}{\partial A}$ denotes $\frac{\partial Y}{\partial AL}L$
		\item Divide both sides by Y(t):
		\[ \frac{\dot{Y}(t)}{Y(t)} = \frac{K(t)}{Y(t)} \frac{\partial Y(t)}{\partial K(t)} \frac{\dot{K}(t)}{K(t)} + \frac{L(t)}{Y(t)} \frac{\partial Y(t)}{\partial L(t)} \frac{\dot{L}(t)}{L(t)} + \frac{A(t)}{Y(t)} \frac{\partial Y(t)}{\partial A(t)} \frac{\dot{A}(t)}{A(t)} \]
	\end{itemize}
\end{frame}

\begin{frame}{Growth Accounting: Basics}
	\[ \frac{\dot{Y}(t)}{Y(t)} = \frac{K(t)}{Y(t)} \frac{\partial Y(t)}{\partial K(t)} \frac{\dot{K}(t)}{K(t)} + \frac{L(t)}{Y(t)} \frac{\partial Y(t)}{\partial L(t)} \frac{\dot{L}(t)}{L(t)} + \frac{A(t)}{Y(t)} \frac{\partial Y(t)}{\partial A(t)} \frac{\dot{A}(t)}{A(t)} \] \vspace{-10pt}
	\begin{itemize}
		\item Elasticity of output with respect to capital and labor
		\[ \alpha_{K}(t) = \frac{K(t)}{Y(t)} \frac{\partial Y(t)}{\partial K(t)} \hspace{30pt} \alpha_{L}(t) = \frac{L(t)}{Y(t)} \frac{\partial Y(t)}{\partial L(t)}  \]
		\item We get:
		\[ \frac{\dot{Y}(t)}{Y(t)} = \alpha_{K}(t) \frac{\dot{K}(t)}{K(t)} + \alpha_{L}(t) \frac{\dot{L}(t)}{L(t)} + R(t) \]
		where 
		\[R(t) = \frac{A(t)}{Y(t)} \frac{\partial Y(t)}{\partial A(t)} \frac{\dot{A}(t)}{A(t)} \]
		is referred to as the Solow Residual 
	\end{itemize}
\end{frame}


\begin{frame}{Growth Accounting: Basics}
	\[ \frac{\dot{Y}(t)}{Y(t)} = \alpha_{K}(t) \frac{\dot{K}(t)}{K(t)} + \alpha_{L}(t) \frac{\dot{L}(t)}{L(t)} + R(t) \] \vspace{-10pt}
	\begin{itemize}
		\itemsep1em 
		\item In principle measurable:
		\begin{itemize}
			\item Growth in output: $\dot{Y}(t)/Y(t)$
			\item Growth in capital: $\dot{K}(t)/K(t)$
			\item Growth in labor: $\dot{L}(t)/L(t)$
			\item Elasticity of output with respect to capital: $\alpha_{K}(t)$
			\item Elasticity of output with respect to labor: $\alpha_{L}(t)$
		\end{itemize}
		\item Yields $R(t)$ as a residual (hence ``Solow residual'' name)
		\begin{itemize}
			\item One perspective: Measure of our ignorance
		\end{itemize}
	\end{itemize}
\end{frame}

\begin{frame}{Measurement: Capital}
	\begin{itemize}
		\itemsep1em 
		\item Ideally we could measure flow of services from capital
		\item In practice: Measure stock and assume flow is proportional to stock
		\item Perpetual inventory method:
		\[K(t+1) = K(t) + I(t) - \delta K(t) \] \vspace{-15pt}
		\begin{itemize}
			\item Start with some $K(0)$
			\item Measure $I(t)$ from National Income and Product Accounts
			\item Use estimates of $\delta$
		\end{itemize} 
	\end{itemize}
\end{frame}

\begin{frame}{Measurement: Quality of Inputs}
	\begin{itemize}
		\itemsep1em
		\item Simple measure of labor input: hours worked
		\item But workers differ, e.g., in education and health
		\item Increase in output may be due to increases in labor quality
		\item Jorgenson and Griliches (1967): 
		\begin{itemize}
			\item Disaggregate inputs by schooling, etc.
			\item Weight each category by average wage
		\end{itemize}
		\item Growth in overall labor input weighted average of categories
		\item Can also be done for capital  
	\end{itemize}
\end{frame}


\begin{frame}{Measurement: Output Elasticities}
	\[ \alpha_{K}(t) = \frac{K(t)}{Y(t)} \frac{\partial Y(t)}{\partial K(t)} \hspace{30pt} \alpha_{L}(t) = \frac{L(t)}{Y(t)} \frac{\partial Y(t)}{\partial L(t)}  \] \vspace{-10pt}
	\begin{itemize}
		\item If labor and capital earn their marginal product:
		\[ r(t) = \frac{\partial Y(t)}{\partial K(t)} \hspace{30pt} w(t) = \frac{\partial Y(t)}{\partial L(t)} \]
		\item In this case output elasticities become factor shares:  
		\[ \alpha_{K}(t) = \frac{r(t)K(t)}{Y(t)} = s_K(t) \hspace{30pt} \alpha_{L}(t) = \frac{w(t)L(t)}{Y(t)} = s_L(t)  \]
		\item Data on factor shares usually used to estimate $\alpha_K(t)$ and $\alpha_L(t)$.
		\item But this is only valid under idealized assumptions \\ {\small (i.e., perfect competition)}
	\end{itemize}
\end{frame}


\begin{frame}{Why Not Estimate?}
	\[ \frac{\dot{Y}(t)}{Y(t)} = \alpha_{K}(t) \frac{\dot{K}(t)}{K(t)} + \alpha_{L}(t) \frac{\dot{L}(t)}{L(t)} + R(t) \] \vspace{-10pt}
	\begin{itemize}
		\itemsep1em 
		\item Alternative approach: Estimate this equation using data on \\ $\dot{Y}(t)/Y(t)$, $\dot{K}(t)/K(t)$, $\dot{L}(t)/L(t)$
		\begin{itemize}
			\item Recover $\alpha_{K}$ and $\alpha_{L}$ as parameters
			\item Recover $R(t)$ as a residual
		\end{itemize}
		\item Why not do this instead? \pause
		\item Would be hard since productivity affects inputs \\ (i.e., labor and capital are endogenous)
	\end{itemize}
\end{frame}


\begin{frame}
	\begin{figure}
		\centering
		\includegraphics[height = 0.90  \textheight]{FiguresTables/Barrosalaimartin2004Table101a.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Barro and Sala-I-Martin (2004) who report results from Christensen, Cummings, Jorgenson (1980) \\ \vspace{-5pt} in panel A and Jorgenson and Yip (2001) in panel B. Not per capita.}
\end{frame}


\begin{frame}
	\begin{figure}
		\centering
		\includegraphics[height = 0.90  \textheight]{FiguresTables/Barrosalaimartin2004Table101b.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Barro and Sala-I-Martin (2004) who report results from Elias (1990) in panel C and Young (1995) \\ \vspace{-5pt} in panel D.}
\end{frame}


\begin{frame}{East Asian Growth Miracle}
	\begin{itemize}
		\itemsep1em 
		\item Young's (1995) results were surprising to many
		\item Could it really be that high growth rates were not associated with \\ large changes in TFP?
		\item Some considered them less miraculous due to this. (But why?)
		\item Hsieh (2002) took a different approach
	\end{itemize}
\end{frame}


\begin{frame}{Dual Growth Accounting: Hsieh (2002)}
	\begin{itemize}
		\item We start with the accounting identity:
		\[Y = rK + wL \]
		\item Take logarithms and differentiate with respect to time:
		\[\frac{\dot{Y}}{Y} = s_K \left(\frac{\dot{r}}{r} + \frac{\dot{K}}{K} \right) + s_L \left(\frac{\dot{w}}{w} + \frac{\dot{L}}{L} \right) \]
		\item Rearrange
		\[ \frac{\dot{Y}}{Y} - s_K \frac{\dot{K}}{K} - s_L \frac{\dot{L}}{L} = s_K \frac{\dot{r}}{r} + s_L \frac{\dot{w}}{w} \] \vspace{-10pt}
		\begin{itemize}
			\itemsep0.5em
			\item LHS: ``primal'' measure of Solow residual (what we had before)
			\item RHS: ``dual'' measure of Solow residual
		\end{itemize}
	\end{itemize}
\end{frame}


\begin{frame}{Dual Growth Accounting: Hsieh (2002)}
	\[ \frac{\dot{Y}}{Y} - s_K \frac{\dot{K}}{K} - s_L \frac{\dot{L}}{L} = s_K \frac{\dot{r}}{r} + s_L \frac{\dot{w}}{w} \] \vspace{-5pt}
	\begin{itemize}
		\itemsep1em 
		\item Primal and dual approach should yield the same answer
		\item If one is (in)valid, the other is (in)valid
		\item Hsieh (2002) applied dual approach to East Asian ``Tigers''
	\end{itemize}
\end{frame}


\begin{frame}{East Asian Growth: Primal vs. Dual}
	\begin{figure}
		\centering
		\includegraphics[height = 0.48  \textheight]{FiguresTables/Barrosalaimartin2004Table102.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Barro and Sala-I-Martin (2004)}
\end{frame}


\begin{frame}{East Asian Growth: Primal vs. Dual}
	Hsieh (2002) argues: \vspace{5pt}
	\begin{itemize}
		\itemsep1em 
		\item NIPA data implies that capital/output ratio rose sharply
		\item Since factor shares are roughly constant, this implies that \\ rate of return on capital should have fallen sharply
		\item True for Korea but not for Singapore
		\item Singapore's NIPA overstate investment
	\end{itemize}
\end{frame}


\begin{frame}{Return on Capital: Korea}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/Hsieh2002Figure1.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Hsieh (2002)}
\end{frame}


\begin{frame}{Return on Capital: Singapore}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/Hsieh2002Figure2.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Hsieh (2002)}
\end{frame}


\begin{frame}{Who is Right? Young or Hsieh?}
	\begin{itemize}
		\itemsep1em 
		\item Subsequent research has tended to favor Young
		\item Fernald and Nieman (2011): \vspace{5pt}
		\begin{itemize}
			\itemsep0.7em 
			\item Economy with two sectors: favored and unfavored
			\item Distortions mean both primal and dual measures of TFP \\ differ form true productivity growth
			\item Bottom-up measurement for Singapore indicates low growth \\ in aggregate technology 
			\item Hsieh's user cost estimates are from unfavored sector
			\item Falling pure profits also missed by Hsieh's approach
		\end{itemize}
	\end{itemize}
\end{frame}


\begin{frame}{Growth Accounting and Sources of Growth}
	\begin{itemize}
		\item Growth accounting is just accounting, not causal analysis
		\item Example: 
		\[ Y = AK^{\alpha}(Le^{xt})^{1-\alpha} \] \vspace{-10pt}
		\begin{itemize}
			\item Suppose $A$ and $L$ are constant
			\item $x$ is labor-augmented growth in technology
		\end{itemize}
		\item Take logarithms and differentiate with respect to time:
		\[ \frac{\dot{Y}}{Y} = \alpha \frac{\dot{K}}{K} + (1-\alpha) x \] 
	\end{itemize}
\end{frame}


\begin{frame}{Growth Accounting and Sources of Growth}
	\[ \frac{\dot{Y}}{Y} = \alpha \frac{\dot{K}}{K} + (1-\alpha) x \] \vspace{-10pt}
	\begin{itemize}
		\item In Solow and Ramsey models: capital-output ratio will be constant along a balanced growth path
		\[ \frac{\dot{Y}}{Y} = \frac{\dot{K}}{K} = x \]
		\item $\alpha x$ of growth attributed to growth of capital
		\item TFP growth measured to be $\hat{g} = (1-\alpha) x$ 
		\item But growth in capital stock is consequence of growth in technology
		\item To attribute to technology both direct and indirect effects on GDP \\ we need to divide measured TFP growth by $(1-\alpha)$
	\end{itemize}
\end{frame}


\begin{frame}
	\begin{figure}
		\centering
		\includegraphics[height = 0.65  \textheight]{FiguresTables/Barrosalaimartin2004Table103.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Barro and Sala-I-Martin (2004)}
\end{frame}


\begin{frame}{Growth Accounting and Sources of Growth}
	\begin{quote}
		A small positive number for $\hat{g}$ is, in principle, consistent with a situation in which technological progress is ultimately responsible for a small part of GDP growth, but it is also consistent with a situation in which it is ultimately responsible for all of GDP growth.
	\end{quote}
	\hspace{2in} Barro and Sala-I-Martin (2004, p. 460)
\end{frame}


\begin{frame}{Alternative Growth Accounting Approach}
	\begin{itemize}
		\item Start with a Cobb-Douglas production function:
		\[Y_t = A_t K_t^{\alpha}H_t^{1-\alpha} \] \vspace{-20pt}
		\begin{itemize}
			\item Here $H_t$ denotes human capital
		\end{itemize}
		\item Divide both sides by $Y_t^{\alpha}$ and raise to power $1/(1-\alpha)$:
		\[ Y_t = \left( \frac{K_t}{Y_t} \right)^{\frac{\alpha}{1-\alpha}} H_t Z_t \]
		where $Z_t = A_t^{\frac{1}{1-\alpha}}$
		\item Divide through by $L_t$:
		\[ \frac{Y_t}{L_t} = \left( \frac{K_t}{Y_t} \right)^{\frac{\alpha}{1-\alpha}} \frac{H_t}{L_t} Z_t \]
	\end{itemize}
\end{frame}


\begin{frame}{Alternative Growth Accounting Approach}
	\[ \frac{Y_t}{L_t} = \left( \frac{K_t}{Y_t} \right)^{\frac{\alpha}{1-\alpha}} \frac{H_t}{L_t} Z_t \] \vspace{-10pt}
	\begin{itemize}
		\itemsep0.5em 
		\item Decomposes per capita (or per hour) growth into:
		\begin{itemize}
			\item Capital deepening: $K_t/Y_t$
			\item Growth in human capital per hour: $H_t/L_t$
			\item Total factor productivity: $Z_t$
		\end{itemize}
		\item Importantly Solow and Ramsey model imply that $K_t/Y_t$ is constant along a balanced growth path
		\item Take logarithms and differentiate with respect to time to get a \\ growth accounting equation
		\item This approach popularized by Klenow and Rodriguez-Clare (1997) \\ {\small (Goes back at least to David (1977))}
	\end{itemize}
\end{frame}


\begin{frame}
	\begin{figure}
		\centering
		\includegraphics[height = 0.6  \textheight]{FiguresTables/Jones2016Table3.pdf}
	\end{figure}
	\vspace{-2mm}
	{\scriptsize Source: Jones (2016)}
\end{frame}

\begin{frame}{Philippon (2023): Additive Growth?}
	\begin{itemize}
		\itemsep1em 
		\item Very new paper! No peer review, back-and-forth, etc. 
		\item Central claim: 
		\begin{itemize}
			\item We usually assume growth is exponential:
			\[ A_{t+\tau} = A_t(1+g)^{\tau} \]
			\item In fact, growth is linear:
			\[ A_{t+\tau} = A_t + b\tau \] 
		\end{itemize}
		\item Data:
		\begin{itemize}
			\item U.S. TFP 1947-2019 from Fernald (2012)
			\item TFP in 23 countries 1890-2019 from Bergeaud, Cette, Lecat (2016)
		\end{itemize}
	\end{itemize}
\end{frame}


\begin{frame}{Exponential versus Linear}
	\begin{itemize}
		\itemsep1em 
		\item Exponential growth:
	 	\[ \begin{array}{ccl}
	 		 A_{t+\tau} = A_t(1+g)^{\tau} & => & \log A_{t+\tau} = \log A_t + \tau \log(1+g) \\
	 		 &  => & \log A_{t+\tau} \approx \log A_t + g \tau
	 	\end{array} \]
		\item Linear growth 
		\[ A_{t+\tau} = A_t + b \tau \]
	\end{itemize}
\end{frame}


\begin{frame}{Philippon (2023): Additive Growth?}
	\begin{itemize}
		\itemsep1em 
		\item Consider the U.S. post-WWII sample
		\item Use the first half of sample to predict $A(t)$ for the second half
		\item Compare prediction of:
		\begin{itemize}
			\item Person who believes in exponential growth (Model G)
			\item Person who believes in linear growth (Model A)
		\end{itemize}
	\end{itemize}
\end{frame}


\begin{frame}{Linear Growth Fits Better Out of Sample}
	\begin{figure}
		\centering
		\includegraphics[height = 0.6  \textheight]{FiguresTables/Philippon2023Figure1.pdf}
	\end{figure}
	\vspace{-2mm}
	{\scriptsize Source: Philippon (2023)}
\end{frame}


\begin{frame}{Linear Growth in the Long-Run}
	\begin{itemize}
		\itemsep1em 
		\item Growth cannot have been linear forever
		\item If so, A(t) would be negative at some point in the past
		\item Philippon proposes that ``General Purpose Technologies'' cause breaks
		\begin{itemize}
			\item Enlightenment / Glorious Revolution?
			\item Steam Engine / Industrial Revolution
			\item Electrification / Second Industrial Revolution
		\end{itemize}
	\end{itemize}
\end{frame}


\begin{frame}{Occasional Breaks over Longer Sample}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/Philippon2023Figure10.pdf}
	\end{figure}
	\vspace{-2mm}
	{\scriptsize Source: Philippon (2023)}
\end{frame}


\begin{frame}{Occasional Breaks over Longer Sample}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/Philippon2023Figure11.pdf}
	\end{figure}
	\vspace{-2mm}
	{\scriptsize Source: Philippon (2023). U.K. Pseudo-TFP (GDP per capita to the power 2/3).}
\end{frame}


\end{document}

