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{ \setbeamertemplate{headline}{}
\frame{

	\vspace{5mm}

	\begin{center}
    	\textcolor{cb}{{\Large Accounting for Cross-Country Income Differences Revisited}}
   	\end{center}

    	\vspace{5mm}

     	\begin{center}
   	\begin{tabular}{cc}\textbf{David Lagakos} ~~~& \textbf{Todd Schoellman} \\[+1ex]
	Boston University ~~~& Minneapolis Fed
	\end{tabular}
    	\end{center}

	\vspace{25mm}
	\textit{{\tiny The views expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System.}}

}
}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}{Motivation}

    \main{The big question: why does output per worker vary so much across countries? }

    \bigskip

    \main{Development accounting helps answer this question}
	\begin{itemize}
	\item Given $F$ and measured $Y_i$, $K_i$, $L_i$ $\Rightarrow$ solve for $A_i$
	\item Use $F$ to evaluate each factor's contribution
    \item \textit{Account}, not \textit{explain}: proximate sources, not root causes
	\end{itemize}

\end{frame}

\begin{frame}{Analogy: Decomposing Mortality Gaps}

    \main{Decompose mortality gaps by cause of death}
	\begin{itemize}
	\item Malaria, diarrheal disease account for much of higher mortality in developing countries
    \item Does not pinpoint the root cause or most effective policies
    \item Useful information for researchers and policy makers
	\end{itemize}

\end{frame}


\begin{frame}{Benchmark: \citet{case05}}

	\main{Inputs account for little; most of development attributed to TFP}
	\begin{itemize}
	\item ``As of now, the answer to the development-accounting question – do observed differences in the factors employed in production explain most of the cross-country variation in income – is: no, way no.''
    \item Needed: A Theory of TFP \citep{pres98}
	\end{itemize}

    \bigskip

    \main{Similar findings pervasive in early literature, some updates}
    \begin{itemize}
    \item \citet{klen97}, \citet{hall99}, \citet{hsie10}, \citet{case16c}
    \end{itemize}

\end{frame}

\begin{frame}{This article: two decades of improved measurement}

	\main{Synthesize the literature since Caselli (2005), with updated calculations}
	\begin{itemize}
	\item Hours: workers in poorer countries work \textit{more} -- deepens the puzzle (-9\%)
	\item Physical capital: still a small role (-5 to 5\%)
	\item Human capital: much larger role (60 to 70\%)
	\end{itemize}

	\bigskip

	\main{Bottom line: inputs account for 45 to 65\% of GDP per worker differences}
	\begin{itemize}
	\item Classic specification with the same data: 28\%
	\item Suggests a shift back toward research on variation in inputs
	\end{itemize}

\end{frame}

\begin{frame}{Outline}

	\begin{enumerate}
	\item \main{Overview of Basic Structure and Facts}
	\item Measures of the Labor Input
	\item Measurement of Physical Capital
	\item The Constructive Approach to Human Capital
	\item Alternative Approaches to Human Capital
	\item Other Advances and Open Areas
    \item Conclusion
	\end{enumerate}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 1. Structure

\begin{frame}{Development accounting structure}

	\main{Common aggregate production function:}
	\begin{align*}
	Y_i = F(A_i, K_i, L_i) \qquad\Rightarrow\qquad Y_i = K_i^{\alpha}\left(A_iL_i\right)^{1-\alpha}
	\end{align*}

    \bigskip

    \main{Cobb-Douglas consistent with stable factor shares}
    \begin{itemize}
        \item Original evidence: $\alpha\approx1/3$ \citep{goll02}
        \item More recent evidence: $\alpha\approx1/2$ \citep{feen15}
        \item We use 1/3, but results are not sensitive under capital-output specification
    \end{itemize}

\end{frame}

\begin{frame}{Standard measures and data sources}

    \main{GDP per worker}
    \begin{itemize}
        \item rgdpo / emp \citep{pwt11}
    \end{itemize}

    \bigskip
    \main {Physical capital}
    \begin{itemize}
        \item Cumulated, depreciated investment expenditures \citep{prit00}
        \item Now constructed and provided as cn \citep{pwt11}
    \end{itemize}

    \bigskip
    \main {Human capital}
    \begin{itemize}
        \item Macro-mincer: $h_i = \exp(0.1\times S_i)$ \citep{bils00}
        \item $S_i$, years of schooling \citep{barr13,edstats}
    \end{itemize}

\end{frame}



\begin{frame}{Two specifications}

	\main{Capital-per-worker specification:}
	\begin{align*}
	y_i = k_i^{\alpha}\left(A_ih_i\right)^{1-\alpha}
	\end{align*}
	\begin{itemize}
	\item Holds $k$ fixed when varying $A$, $h$ -- direct, but implies large changes in $MPK$
	\end{itemize}

	\bigskip

	\main{Capital-output specification:}
	\begin{align*}
	y_i = \left(\frac{K_i}{Y_i}\right)^{\frac{\alpha}{1-\alpha}} h_i A_i
	\end{align*}
	\begin{itemize}
	\item Holds $MPK$ fixed -- captures long-run capital adjustment in a wide class of models
	\end{itemize}

\end{frame}

\begin{frame}{Covariance Metric}

	\main{\citet{klen97} covariance metric:}
	\begin{align*}
	share_x \equiv \frac{\text{cov}\left(\log(x),\log(y)\right)}{\text{var}\left(\log(y)\right)}
	\end{align*}

	\bigskip

	\main{Benefits:}
	\begin{itemize}
	\item Additive and order invariant (useful for combining results across studies)
	\item Rewards factors that both \textit{vary} and \textit{correlate} with $y$
    \item Easy to implement: regress $\log(x)$ on $\log(y)$
	\end{itemize}

\end{frame}


\begin{frame}{The World Income Distribution: Summary Statistics}

    \begin{table}[t]
    
    \setlength {\tabcolsep}{1mm}
    \renewcommand{\arraystretch}{1.2}
    \begin{center}
    
    \begin{tabular}{lcc}
    
    \toprule
    
     & GDP per capita & GDP per worker \\
    
    \midrule
    10th percentile & 2,835 & 7,596 \\
    
    Median & 17,784 & 42,796 \\
    
    Mean & 27,740 & 58,197 \\
    
    90th percentile & 62,891 & 121,293 \\
    
    \midrule
    
    var[log(y)] & 1.40 & 1.06 \\
    
    90-10 ratio & 22.2 & 16.0 \\
    \bottomrule
    
    \end{tabular}
    
    {\setstretch{1}
    \parbox[c]{5.4in}{\vspace{0.1cm}
    {\footnotesize
    \textit{Note}: GDP is measured in 2021 international dollars. The sample includes 153 countries for which data are available on GDP, physical capital, and years of schooling. Source: \citet{pwt11}.
    }}}
    
    \end{center}
    \vspace{-0.1cm}
    \end{table}


\end{frame}

\begin{frame}{Baseline Development Accounting Inputs: Physical Capital}

	\begin{figure}
		\subfloat[Capital per worker]{\includegraphics[width=0.46\textwidth]{Graphs/log_k_log_y_cn.pdf}}
		\qquad
		\subfloat[Capital-output ratio]{\includegraphics[width=0.46\textwidth]{Graphs/K_Y_log_y_cn.pdf}}
	\end{figure}

	\begin{itemize}
	\item Slope of 1.11 for $K/N$; just 0.11 for $K/Y$
	\end{itemize}

\end{frame}

\begin{frame}{Baseline Development Accounting Inputs: Human Capital and TFP}

	\begin{figure}
		\subfloat[Human capital per worker]{\includegraphics[width=0.46\textwidth]{Graphs/hc0_basic_cn.pdf}}
		\qquad
		\subfloat[TFP (US $=$ 1)]{\includegraphics[width=0.46\textwidth]{Graphs/TFP0_cn.pdf}}
	\end{figure}

	\begin{itemize}
	\item Classic accounting relies heavily on large, mysterious TFP gaps (slope 0.48)
	\end{itemize}

\end{frame}

\begin{frame}{Success of Factors, Baseline Measures}

    \begin{table}[t]
    
    \setlength {\tabcolsep}{1mm}
    \renewcommand{\arraystretch}{1.2}
    \begin{center}
    
    \begin{tabular}{lccc}
    
    \toprule
    
    Factor & share & std. err. & \# countries \\
    
    \midrule
    
    \multicolumn{4}{l}{\emph{Capital-per-worker specification}} \\
    
    Physical capital, \(k^\alpha\) & .370 & .012 & 153 \\
    
    Human capital, \(h^{1-\alpha}\) & .152 & .009 & 153 \\
    
    Human + physical capital & .522 & .017 & 153 \\
    
    \midrule
    
    \multicolumn{4}{l}{\emph{Capital-output specification}} \\
    
    Physical capital, \((K/Y)^{\alpha/(1-\alpha)}\) & .055 & .018 & 153 \\
    
    Human capital, \(h\) & .228 & .013 & 153 \\
    
    Human + physical capital & .283 & .025 & 153 \\
    
    \bottomrule
    
    \end{tabular}
    
    {\setstretch{1}
    \parbox[c]{5.4in}{\vspace{0.1cm}
    {\footnotesize
    \textit{Note}: Source: \citet{pwt11} and  \citet{barr13} augmented with \citet{edstats}.
    }}}
    
    \end{center}
    \vspace{-0.1cm}
    \end{table}


\end{frame}

\begin{frame}{Success of Factors, Caselli Metrics}

    \main{Caselli's metrics roughly agree with $K/Y$ specification}

    \begin{table}[t]
    
    \setlength {\tabcolsep}{1mm}
    \renewcommand{\arraystretch}{1.2}
    \begin{center}
    
    \begin{tabular}{lccc}
    
    \toprule
    
    Data & \(\frac{\text{var}[\log(x)]}{\text{var}[\log(y)]}\) & \(\frac{x_{90}/x_{10}}{y_{90}/y_{10}}\) & \# countries \\
    
    \midrule
    
    Caselli's data & 0.39 & 0.34 & 94 \\
    
    Latest data & 0.31 & 0.29 & 153 \\
    \bottomrule
    
    \end{tabular}
    
    \vspace{0.25cm}
    
    \begin{minipage}{0.95\textwidth}
    \footnotesize
    \textit{Note}: Results are based on the metrics proposed by Caselli, which he labels \(success_1\) and \(success_2\). Source: first row: \citet{case05}; second row: data from \citet{pwt11} and \citet{barr13} augmented with \citet{edstats}. 
    \end{minipage}
    
    \end{center}
    \vspace{-0.1cm}
    \end{table}


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 2. Labor

\begin{frame}{Outline}

	\begin{enumerate}
	\item Overview of Basic Structure and Facts
	\item \main{Measures of the Labor Input}
	\item Measurement of Physical Capital
	\item The Constructive Approach to Human Capital
	\item Alternative Approaches to Human Capital
	\item Other Advances and Open Areas
    \item Conclusion
	\end{enumerate}

\end{frame}

\begin{frame}{Labor Input and Development}

    \main{Do people in poor countries work less?}
    \begin{itemize}
    \item Yes: gaps in GDP per capita $>$ gaps in GDP per worker
    \begin{itemize}
        \item $\implies$ developed countries have more workers per capita
    \end{itemize}
    \end{itemize}

    \bigskip

    \main{Due entirely to higher share of children in developing countries}
    \begin{itemize}
        \item Adult employment rates higher in developing countries
        \item Share of children not useful as a proximate explanation, reinforces focus on GDP per worker
    \end{itemize}

\end{frame}

\begin{frame}{Labor Inputs and Development: Workers per Capita}

	\begin{figure}
		\subfloat[Employment-to-population]{\includegraphics[width=0.46\textwidth]{Graphs/emp_pop_sh.pdf}}
		\qquad
		\subfloat[Adult employment rates]{\includegraphics[width=0.46\textwidth]{Graphs/er_a_all.pdf}}
	\end{figure}

\end{frame}

\begin{frame}{New Data: Hours per Worker}

    \main{Wider availability of high-quality labor force surveys with data on hours worked}
    \begin{itemize}
        \item \citet{bick18}: collect and harmonize data from 80 countries
        \item \citet{gesa25}: 160 countries
    \end{itemize}

    \bigskip

    \main{Main finding: more hours per worker in developing countries}
    \begin{itemize}
        \item \emph{Worsens} the development puzzle
    \end{itemize}

\end{frame}


\begin{frame}{Labor Input and Development: Hours per Adult}

	\begin{figure}
		\subfloat[\citet{bick18}]{\includegraphics[width=0.46\textwidth]{Graphs/hwp_a_all_bick.pdf}}
		\qquad
		\subfloat[\citet{gesa25}]{\includegraphics[width=0.46\textwidth]{Graphs/hwp_a_all_gethin_saez.pdf}}
	\end{figure}

\end{frame}

\begin{frame}{Success of Factors Revisited: Labor}

    \main{Puzzle is worse than previously thought}

    \bigskip

    \begin{table}[t]
    
    \setlength {\tabcolsep}{1mm}
    \renewcommand{\arraystretch}{1.2}
    \begin{center}
    
    \begin{tabular}{lccc}
    
    \toprule
    
    Factor & share & std. err. & \# countries \\
    
    \midrule
    
    \multicolumn{4}{l}{\emph{Capital-per-worker specification}} \\
    
    Hours per worker, \(n^{1-\alpha}\) & -.057 & .011 & 139 \\
    
    \midrule
    
    \multicolumn{4}{l}{\emph{Capital-output specification}} \\
    
    Hours per worker, \(n\) & -.086 & .017 & 139 \\
    
    \bottomrule
    
    \end{tabular}
    
    {\setstretch{1}
    \parbox[c]{5.4in}{\vspace{0.1cm}
    {\footnotesize
    \textit{Note}: Source: \citet{pwt11} and \citet{gesa25}.
    }}}
    
    \end{center}
    \vspace{-0.1cm}
    \end{table}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 3. Physical capital

\begin{frame}{Outline}

	\begin{enumerate}
	\item Overview of Basic Structure and Facts
	\item Measures of the Labor Input
	\item \main{Measurement of Physical Capital}
	\item The Constructive Approach to Human Capital
	\item Alternative Approaches to Human Capital
	\item Other Advances and Open Areas
    \item Conclusion
	\end{enumerate}

\end{frame}

\begin{frame}{Overview: What's New?}

    \main{Classic measure: aggregate capital stock}
    \begin{itemize}
        \item Constructed from past investment expenditures
    \end{itemize}

    \bigskip

    \main{Four areas of research:}
    \begin{enumerate}
    \item Capital services
    \item Natural vs. reproducible capital
    \item Public capital
    \item Intangibles
    \end{enumerate}
    
\end{frame}

\begin{frame}{Capital Services: A Primer}

    \main{Capital's contribution to output is the flow of services it provides, $s_t$}
    \begin{itemize}
        \item The use of roads, factories, and equipment for the given period of time
        \item Roughly, their rental price
    \end{itemize}

    \bigskip

    \main{Traditional approach uses instead the total value of the capital stock, $p_t$}
    \begin{itemize}
        \item Cumulated, depreciated investment expenditures
    \end{itemize}

\end{frame}

\begin{frame}{Linking Capital Services and Stocks}

	\main{No-arbitrage equation links $s_t$ to $p_t$, required return on capital $r$, depreciation rate $\delta$:}
	\begin{align*}
	s_t = rp_t + \delta p_{t+1} - (p_{t+1}-p_t)
    \end{align*}

    \bigskip

    \main{If $p_t=p_{t+1}$, then $s_t = p_t(r+\delta)$}
	\begin{itemize}
	\item Using stocks innocuous if $r$, $\delta$ similar across countries
	\end{itemize}

	\bigskip

	\main{\citet{inkl19}: neither holds}
	\begin{itemize}
	\item Developing countries: higher required returns, lower depreciation rates (composition)
	\item Depreciation effect slightly dominates in practice
    \item Now available in PWT (ck)
	\end{itemize}

\end{frame}

\begin{frame}{Capital Services and Development}

	\begin{figure}
		\subfloat[Capital services per worker]{\includegraphics[width=0.46\textwidth]{Graphs/log_k_log_y_ck.pdf}}
		\qquad
		\subfloat[Capital services-output ratio]{\includegraphics[width=0.46\textwidth]{Graphs/K_Y_log_y_ck.pdf}}
	\end{figure}

\end{frame}

\begin{frame}{Defining Natural Capital}

	\main{Natural capital: land and its naturally occurring, economically relevant features}
	\begin{itemize}
	\item Subsoil resources (petroleum, minerals), forests, cropland, pasture
	\item Excluded: urban land, scenic and tourism land (beaches)
	\end{itemize}

	\bigskip

    \main{Excluded from PWT reproducible capital}
    \begin{itemize}
        \item Not the result of past investment expenditures
        \item Contribution of good cropland, subsoil petroleum is distinct from tractors, oil rigs
        \item Could low natural capital account for low productivity?
    \end{itemize}

    \bigskip

    \main{Data on natural resource rents from World Bank make progress possible}
    \begin{itemize}
        \item \citet{mong21}, building on \citet{case07}
    \end{itemize}

\end{frame}

\begin{frame}{Natural capital is not scarce in poor countries}

	\begin{figure}
		\subfloat[Natural resource rents]{\includegraphics[width=0.46\textwidth]{Graphs/natural_share_all_countries.pdf}}
		\qquad
		\subfloat[Natural capital stocks]{\includegraphics[width=0.46\textwidth]{Graphs/log_KN_N_nat_log_y.pdf}}
	\end{figure}

	\begin{itemize}
	\item Data from \citet{mong21}; oil exporters in red
	\end{itemize}

\end{frame}

\begin{frame}{Success of Factors Revisited: Physical Capital}

    \begin{table}[t!]
    
    \normalsize
    \centering
                      
    \begin{tabular}{lccc}
    
    \toprule
    
    Factor & share & std. err. & \# countries \\
    
    \midrule
    
    \multicolumn{4}{l}{\emph{Capital-per-worker specification}} \\
    
    Physical capital stock (cn) & .370 & .012 & 153 \\
    
    Physical capital services (ck) & .367 & .011 & 133 \\
    
    Reproducible capital & .281 & .009 & 128 \\
    
    Natural capital & .015 & .014 & 128 \\
    
    \midrule
    
    \multicolumn{4}{l}{\emph{Capital-output specification}} \\
    
    Physical capital stock (cn) & .055 & .018 & 153 \\
    
    Physical capital services (ck) & .051 & .016 & 133 \\
    
    Reproducible capital & .034 & .014 & 128 \\
    
    Natural capital & -.089 & .021 & 128 \\
    
    \bottomrule
    
    \end{tabular}
    
    \vspace{0.25cm}
    
    \begin{minipage}{0.95\textwidth}
    \footnotesize
    \textit{Note}: Source: \citet{pwt11} for physical capital stock (cn), physical capital services (ck), reproducible capital, and GDP per worker, and \citet{mong21} for natural capital.
    \end{minipage}
    
    \end{table}


\end{frame}

\begin{frame}{What is public capital?}

	\main{Public vs. private capital}
	\begin{itemize}
	\item Public: roads, ports, courts, police stations
	\item Private: factories, trucks
	\item Summing them into ``aggregate $K$'' has no clear meaning $\Rightarrow$ separate, complementary inputs
	\end{itemize}

	\bigskip

	\main{Efficiency of public investment, $\gamma_i$}
	\begin{itemize}
	\item Bridges to nowhere, unfinished school buildings, funds siphoned off
	\item \$1 of public investment yields $\gamma_i < 1$ of usable public capital
	\end{itemize}

\end{frame}

\begin{frame}{Public capital}

    \main{Public capital: capital financed by public investment expenditures}
    \begin{itemize}
        \item Raises two distinct concerns
    \end{itemize}

    \bigskip

    \main{Concern 1: public spending may translate poorly into capital \citep{prit00}}
	\begin{align*}
	K^G_{i,t+1} = K^G_{i,t}(1-\delta^G) + \textcolor{cc}{\gamma_i} I^G_{i,t}
	\end{align*}
    \vspace{-5mm}
	\begin{itemize}
	\item New data: public investment, capital stocks by country \citep{cuba20,imfinvcap21}
	\item Efficiency measures $\gamma_i$ \citep{devadas2018assessing,kamo23} 
	\end{itemize}

	\bigskip
	\main{Concern 2: Public capital may be a separate, complementary input \citep{barr90}}
	\begin{align*}
	Y_i = (K^G_i)^{\zeta}(K^P_i)^{\alpha}\left(A_iL_i\right)^{1-\alpha}
	\end{align*}

\end{frame}

\begin{frame}{Public Capital and Public Investment Efficiency}

	\begin{figure}
		\subfloat[Aggregate Public Capital\\ \citet{cuba20}]{\includegraphics[width=0.46\textwidth]{Graphs/log_KG_aggregate_log_y.pdf}}
		\qquad
		\subfloat[Public Investment Efficiency \\ \citet{devadas2018assessing}]{\includegraphics[width=0.46\textwidth]{Graphs/gamma_wb_iei_log_y.pdf}}
	\end{figure}

	\begin{itemize}
	\item Large gaps in public capital; half of public investment in poorest countries wasted
	\end{itemize}

\end{frame}

\begin{frame}{Success of Factors Revisited: Public and Private Capital}

    \main{Uncertainty about functional forms and parameters, but potentially important}

    \begin{table}[t!]
    
    \normalsize
    \centering
                      
                      \begin{tabular}{lccc}
    
    \toprule
    
    Factor & share & std. err. & \# countries \\
    
    \midrule
    
    \multicolumn{4}{l}{\emph{Capital-per-worker specification}} \\
    
    Private capital, \((k_i^P)^\alpha\) & .404 & .012 & 141 \\
    
    Public capital, \((K_i^G)^{\zeta}\), \(\zeta=0.10\) & .082 & .014 & 141 \\
    
    Public capital, \((\gamma_i K_i^G)^{\zeta}\), \(\zeta=0.10\) & .094 & .014 & 141 \\
    
    Public capital, \((\gamma_i K_i^G)^{\zeta}\), \(\zeta=0.20\) & .189 & .027 & 141 \\
    
    \midrule
    
    \multicolumn{4}{l}{\emph{Capital-output specification}} \\
    
    Private capital, \((K_i^P/Y_i)^{\alpha/(1-\alpha)}\) & .106 & .018 & 141 \\
    
    Public capital, \((K_i^G)^{\zeta/(1-\alpha)}\), \(\zeta=0.10\) & .122 & .020 & 141 \\
    
    Public capital, \((\gamma_i K_i^G)^{\zeta/(1-\alpha)}\), \(\zeta=0.10\) & .142 & .021 & 141 \\
    
    Public capital, \((\gamma_i K_i^G)^{\zeta/(1-\alpha)}\), \(\zeta=0.20\) & .283 & .041 & 141 \\
    
    \bottomrule
    
    \end{tabular}
    
    \begin{minipage}{0.95\textwidth}
    \footnotesize
    \textit{Note}: Source: \citet{cuba20} for public capital and \citet{pwt11} for private capital and output per worker. 
    \end{minipage}
    
    \end{table}
    

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Intangible Capital}

    \main{Intangible capital: long-lived, non-physical assets or resources that enhance productivity}
    \begin{itemize}
        \item R\&D, brand equity, organizational know-how
    \end{itemize}

    \bigskip

    \main{U.S. Time Series: important contributor to growth}
    \begin{itemize}
        \item As important as tangible capital growth \citep{corr09}
    \end{itemize}

    \bigskip

    \main{Currently missing from international investment expenditures, capital stocks}
    \begin{itemize}
        \item $\implies$ included in TFP
        \item More complete uptake of SNA 2008 important for systematic progress
    \end{itemize}

\end{frame}

% 4. Human capital: constructive

\begin{frame}{Outline}

	\begin{enumerate}
	\item Overview of Basic Structure and Facts
	\item Measures of the Labor Input
	\item Measurement of Physical Capital
	\item \main{The Constructive Approach to Human Capital}
	\item Alternative Approaches to Human Capital
	\item Other Advances and Open Areas
    \item Conclusion
	\end{enumerate}

\end{frame}

\begin{frame}{Constructive Approach: Overview}

    \main{Classic measure equates human capital with years of schooling}
    \begin{itemize}
        \item $h_i = \exp(0.1 \times S_i)$
    \end{itemize}

    \bigskip

    \main{One strand of research adds additional dimensions of human capital}
    \begin{itemize}
        \item Education quality
        \item Experience
        \item Early childhood, parents, and culture
        \item Health
    \end{itemize}

    \bigskip

    \main{These dimensions are summed to \emph{construct} a new, more expansive measure of human capital}


\end{frame}

\begin{frame}{Education Quality}

    \main{Human capital generated per year of schooling}

    \bigskip

    \main{Three forms of evidence}
    \begin{enumerate}
        \item Cross-country test scores: Harmonized Learning Outcomes \citep{angr21}
        \item Rate of test score improvement \citep{kaar14}
        \item Returns to schooling of foreign-educated migrants \citep{scho12}
    \end{enumerate}
   

\end{frame}

\begin{frame}{Education Quality Measures}

	\begin{figure}
		\subfloat[Average HLO Score \\ \citet{angr21}]{\includegraphics[width=0.46\textwidth]{Graphs/z_hlo.pdf}}
        \qquad
		\subfloat[Immigrant Returns to Schooling \\ Following \citet{scho12}]{\includegraphics[width=0.46\textwidth]{Graphs/returns_migrants.pdf}}
	\end{figure}

\end{frame}

\begin{frame}{Valuing education quality}

	\main{Extend the macro-Mincer approach:}
	\begin{align*}
	h_i = \exp(0.1\times S_i + \rho Q_i), \qquad \rho = 0.20 \text{  \citep{angr21}}
	\end{align*}

	\bigskip

	\main{Or let quantity and quality interact \citep{scho12}}
	\begin{align*}
	h_i = \exp\left[\frac{(S_iQ_i)^{\eta}}{\eta}\right], \qquad \eta \approx 0.5
	\end{align*}

 \end{frame}

\begin{frame}{Experience}

    \main{Human capital accumulated through work}
    \begin{itemize}
        \item Typically proxied as potential experience = years since graduation $\approx$ age$-s-6$
    \end{itemize}

    \bigskip

    \main{Early literature: no substantive differences}
    \begin{itemize}
        \item Level of experience is similar: developing countries have younger workers, less schooling
        \item Returns to experience are similar
    \end{itemize}

    \bigskip

    \main{Revision is about better data}
    \begin{itemize}
        \item Returns to experience higher in developed countries \citep{laga18,jedw23}
        \item Returns to experience are portable, probably reflect human capital \citep{laga18b}
        \item Potential mechanism: less training in developing countries \citep{ma24}
    \end{itemize}

\end{frame}

\begin{frame}{Returns to Potential Experience}

	\begin{figure}
		\subfloat[Immigrants\\ \citet{laga18b}]{\includegraphics[width=0.46\textwidth]{Graphs/retexp_lagakos.pdf}}
		\qquad
		\subfloat[Non-migrants \\ \citet{jedw23}]{\includegraphics[width=0.46\textwidth]{Graphs/retexp_jedwab.pdf}}
	\end{figure}

\end{frame}

\begin{frame}{Early Childhood, Parents, and Culture}

    \main{Early childhood environment}
    \begin{itemize}
        \item Refugees: no effect on adult outcomes from an additional year of early childhood abroad \citep{scho16}
    \end{itemize}

    \bigskip

    \main{Parents}
    \begin{itemize}
        \item Large gaps in test scores for children at start of school \citep{sing20}
        \item Second-generation immigrants ``inherit'' parental-origin test scores; fades with time \citep{deph21}
    \end{itemize}

    \bigskip

    \main{Culture}
    \begin{itemize}
        \item Migrants from richer countries are more productive in Sweden \citep{ek24}
        \item Mechanism: migrants from richer countries have culture of autonomy, sort to non-routine jobs
    \end{itemize}

\end{frame}

\begin{frame}{Health}

	\main{Health: height as broad proxy} 
	\begin{itemize}
	\item 3.4\% return per cm; richest and poorest countries differ by up to 15 cm \citep{weil07}
	\item Little new work on aggregate health stocks since -- open area
	\end{itemize}

\end{frame}

\begin{frame}{Summing Up}

    \main{Goal: add dimensions to construct total human capital}

    \bigskip

    \main{Concern: double-counting}
    \begin{itemize}
        \item Test score differences reflect parents and culture, not just school systems \citep{sing20,deph21}
        \item Health affects years of schooling \citep{weil07}
    \end{itemize}

    \bigskip

    \main{Approach}
    \begin{itemize}
        \item Add years of schooling, test scores, experience, health
        \item Each constructed to limit double-counting
        \item Development accounting not well-suited to identify causal effects
    \end{itemize}

\end{frame}

\begin{frame}{Human Capital Stocks Revisited}
    \vspace{-2mm}
	\begin{figure}
		\subfloat[Years of schooling]{\includegraphics[width=0.35\textwidth]{Graphs/hc0.pdf}}
		\subfloat[$+$ quality (test scores)]{\includegraphics[width=0.35\textwidth]{Graphs/hc1c.pdf}}\\
		\subfloat[Experience (non-migrants)]{\includegraphics[width=0.35\textwidth]{Graphs/hc2.pdf}}
		\subfloat[Health]{\includegraphics[width=0.35\textwidth]{Graphs/hc_health.pdf}}
	\end{figure}

\end{frame}

\begin{frame}{Total human capital}

	\begin{figure}
		\includegraphics[width=0.65\textwidth]{Graphs/hc3c_health.pdf}
	\end{figure}

\end{frame}

\begin{frame}{Success of Factors Revisited: Human Capital}

\begin{table}[t!]
\centering
\begin{adjustbox}{max totalheight=0.8\textheight, max width=\linewidth}
\begin{tabular}{lccc}
\toprule
Factor, \(x\) & share & std. err. & \# countries \\
\midrule
\multicolumn{4}{l}{\emph{Capital-per-worker specification}} \\
Years of schooling & .152 & .009 & 153 \\
\quad ... + schooling quality (test scores) & .240 & .012 & 142 \\
\quad ... + schooling quality (immigrants) & .238 & .019 & 123 \\
Experience (non-migrants) & .102 & .017 & 109 \\
Experience (immigrants) & .153 & .033 & 65 \\
Health & .063 & .006 & 152 \\
Human capital (years + quality + experience + health) & .432 & .025 & 106 \\
\midrule
\multicolumn{4}{l}{\emph{Capital-output specification}} \\
Years of schooling & .228 & .013 & 153 \\
\quad ... + schooling quality (test scores) & .360 & .018 & 142 \\
\quad ... + schooling quality (immigrants) & .357 & .029 & 123 \\
Experience (non-migrants) & .153 & .025 & 109 \\
Experience (immigrants) & .230 & .049 & 65 \\
Health & .095 & .009 & 152 \\
Human capital (years + quality + experience + health) & .648 & .037 & 106 \\
\bottomrule
\end{tabular}
\end{adjustbox}%
\\[0.15em]
\parbox{\linewidth}{%
\fontsize{6}{6.8}\selectfont
\textit{Note}: Sources, by row: (1): \citet{barr13} augmented with \citet{edstats}; (2): Harmonized Learning Outcomes from \citet{angr21}, valued at 20 percent per standard deviation; (3): own estimates following \citet{scho12}; (4): \citet{jedw23}; (5): \citet{laga18b}; (6): height at age 19 from \citet{ncdrisc20,ncdrisc_data}, converted into human capital following \citet{weil07}. Throughout, GDP per worker is taken from \citet{pwt11}.%
}

\end{table}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 5. Deductive + imperfect substitutes

\begin{frame}{Outline}

	\begin{enumerate}
	\item Overview of Basic Structure and Facts
	\item Measures of the Labor Input
	\item Measurement of Physical Capital
	\item The Constructive Approach to Human Capital
	\item \main{Alternative Approaches to Human Capital}
	\item Other Advances and Open Areas
    \item Conclusion
	\end{enumerate}

\end{frame}

\begin{frame}{Other Approaches: Overview}

    \main{Constructive approach: expands on classic measurement}
    \begin{itemize}
        \item Adds additional dimensions
    \end{itemize}

    \bigskip

    \main{Other approaches}
    \begin{itemize}
        \item Deductive approach: wage changes of migrants
        \item Imperfect substitution: incorporates canonical model from labor economics
    \end{itemize}
    
\end{frame}

\begin{frame}{Deductive Approach}

    \main{Cross-country migrants convey useful information \citep{hend02}}
    \begin{itemize}
        \item Carry their portable human capital to countries with new physical capital, TFP
        \item Challenge: selection, skill transfer, discrimination
    \end{itemize}

    \bigskip

    \main{New strategy: wage changes of migrants \citep{hend18}}
    \begin{itemize}
        \item Small wage change $\implies$ small role for country $(K/Y,A)$ $\implies$ large role for $h$
        \item Large wage change $\implies$ large role for country $\implies$ small role for $h$
        \item Same migrant in two countries reduces concerns about selection
    \end{itemize}

    \bigskip

    \main{Deduce the importance of human capital}

\end{frame}

\begin{frame}{Human Capital, Deductive Approach}

	\begin{figure}
		\includegraphics[width=0.46\textwidth]{Graphs/wage_gains.pdf}
	\end{figure}

    \main{Poorest countries: wage gains ($3.2\times$) are small relative to GDP per worker gaps ($31.8\times$)}
    \begin{itemize}
        \item Human capital accounts for $1-\frac{\log(3.2)}{\log(31.8)}$ = 66 percent
        \item Similar to constructive approach; see also \citet{mart24}
    \end{itemize}

\end{frame}

\begin{frame}{Imperfect Substitutes Approach}

	\main{Canonical model of labor economics \citep{katz92}}
	\begin{align*}
	L_i = \left(z_{u,i}L_{u,i}^{\frac{\sigma-1}{\sigma}} + z_{s,i}L_{s,i}^{\frac{\sigma-1}{\sigma}}\right)^{\frac{\sigma}{\sigma-1}}
	\end{align*}

    \bigskip

    \main{Supply of each type of labor:}
    \begin{align*}
        L_{u,i}=N_{u,i}h_{u,i}\\
        L_{s,i}=N_{s,i}h_{s,i}
    \end{align*}

    \bigskip

    \main{Elasticity of substitution $\sigma \in [1,2]$}
    \begin{itemize}
        \item Relative supply, skill bias of technology matter \citep{gold08}
    \end{itemize}

\end{frame}

\begin{frame}{Relative Skill Efficiency}

	\main{Observed skill premium:}
	\begin{align*}
	\omega_i \equiv \frac{w_{s,i}h_{s,i}}{w_{u,i}h_{u,i}} = \frac{z_{s,i}}{z_{u,i}}\left(\frac{h_{s,i}}{h_{u,i}}\right)^{\frac{\sigma-1}{\sigma}}\left(\frac{N_{s,i}}{N_{u,i}}\right)^{-\frac{1}{\sigma}}
	\end{align*}

    \bigskip

    \main{Developing countries have much lower relative supply of skilled labor}
    \begin{itemize}
        \item Force towards much higher values of $\omega_i$
        \item Available evidence: $\omega_i$ is flat or slightly declining with development \citep{bane05,ross22}
    \end{itemize}

    \bigskip

    \main{Developed countries must have much higher relative efficiency of skilled labor \citep{ross22}}
    \begin{align*}
         \frac{z_{s,i}}{z_{u,i}}\left(\frac{h_{s,i}}{h_{u,i}}\right)^{\frac{\sigma-1}{\sigma}}
    \end{align*}

\end{frame}


\begin{frame}{Relative Skill Efficiency}

	\begin{figure}
		\includegraphics[width=0.46\textwidth]{Graphs/log_rel_sk_eff.pdf}
	\end{figure}

	\begin{itemize}
	\item 16$\times$ higher in developed countries ($\sigma = 1.5$)
	\end{itemize}

\end{frame}

\begin{frame}{Sources of Relative Skill Efficiency}

    \main{Skill efficiency captures two effects}
    \begin{itemize}
        \item Relative human capital of skilled workers $h_{s,i}/h_{u,i}$
        \item Skill bias of technology $z_{s,i}/z_{u,i}$
        \item Disagreement about which matters quantitatively \citep{case06,jone14}
    \end{itemize}

    \bigskip

    \main{Additional data needed}
    \begin{itemize}
        \item Returns to schooling of immigrants
        \item Trade data
    \end{itemize}

    \bigskip

    \main{Result: Majority (67--91 percent) due to skill bias of technology}
    
\end{frame}

\begin{frame}{Imperfect Substitution and Development Accounting}

    \main{Imperfect substitutes $\implies$ labor aggregator may matter much more}
    \begin{itemize}
        \item Simple example: assume $z_{u,i}=h_{u,i} \equiv 1$
    \end{itemize}

    \bigskip

    \main{Express labor aggregator in terms of observables:}
	\begin{align*}
	L_i = N_{u,i}^{\frac{1}{1-\sigma}}\left(N_{u,i} + \omega_iN_{s,i}\right)^{\frac{\sigma}{\sigma-1}}
	\end{align*}
    \vspace{-5mm}
    \begin{itemize}
        \item Mixes quantity, quality of skilled labor \& skill bias of technology \citep{case13}
    \end{itemize}

\end{frame}

\begin{frame}{Importance of Imperfect Substitutes Labor Aggregator}

    \main{Potentially important accounting implications, sensitive to $\sigma$}

    \begin{table}[t]
    
    \setlength {\tabcolsep}{1mm}
    \renewcommand{\arraystretch}{1.2}
    \begin{center}
    
    \begin{tabular}{lccc}
    
    \toprule
    
    Factor & share & std. err. & \# countries \\
    
    \midrule
    
    \multicolumn{4}{l}{\emph{Capital-per-worker specification}} \\
    
    Aggregator, \(\sigma=1.2\) & .659 & .057 & 140 \\
    
    Aggregator, \(\sigma=1.5\) & .340 & .026 & 140 \\
    
    Aggregator, \(\sigma=2.0\) & .234 & .017 & 140 \\
    
    \midrule
    
    \multicolumn{4}{l}{\emph{Capital-output specification}} \\
    
    Aggregator, \(\sigma=1.2\) & .987 & .085 & 140 \\
    
    Aggregator, \(\sigma=1.5\) & .510 & .040 & 140 \\
    
    Aggregator, \(\sigma=2.0\) & .352 & .025 & 140 \\
    
    \bottomrule
    
    \end{tabular}
    
    {\setstretch{1}
    \parbox[c]{5.4in}{\vspace{0.1cm}
    {\footnotesize
    \textit{Note:} Source: own calculation using educational attainment data from \citet{barr13} augmented with \citet{edstats}; \citet{pwt11}.
    }}}
    
    \end{center}
    \end{table}
    

\end{frame}

\begin{frame}{Correlation Suggests Endogenous Mechanism}

    \main{High correlation among skilled labor share, skill bias of technology, human capital of skilled worker}
    \begin{itemize}
        \item Suggests endogenous response or co-determination
        \item Previous slide: can be quantitatively important in aggregate
    \end{itemize}

    \bigskip

    \main{Similar idea may apply to other dimensions of human capital}
    \begin{itemize}
        \item Complementarity between skill and culture or experience human capital? 
    \end{itemize}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 6. Other advances + conclusion

\begin{frame}{Outline}

	\begin{enumerate}
	\item Overview of Basic Structure and Facts
	\item Measures of the Labor Input
	\item Measurement of Physical Capital
	\item The Constructive Approach to Human Capital
	\item Alternative Approaches to Human Capital
	\item \main{Other Advances and Open Areas}
    \item Conclusion
	\end{enumerate}

\end{frame}

\begin{frame}{Other advances}

	\main{Capital-labor substitutability}
	\begin{itemize}
    \item \citet{case05}: capital \& labor account for more if they are complements
    \item Recent plant-level estimates strongly suggest elasticity $<$ 1 \citep{ober21}
    \item Aggregate elasticity likely larger; value uncertain 
	\end{itemize}

	\bigskip

	\main{Sectors}
	\begin{itemize}
    \item \citet{case05}, \citet{rest08}: Agriculture is uniquely unproductive
        \begin{itemize}
            \item $\frac{\text{90-10 ratio in ag}}{\text{90-10 ratio in non-ag}}=\frac{43.5}{4.5}=9.7$
        \end{itemize}
    \item New GGDC data on sectoral PPPs reduce the gap \citep{bopp23}
        \begin{itemize}
            \item  $\frac{38.5}{11.9}=3.2$ 
        \end{itemize}
    \item Data point to CES production function in ag \citep{chen20,bopp23}
	\end{itemize}


\end{frame}

\begin{frame}{Other advances}

	\main{Firm productivity}
    \begin{itemize}
        \item Management quality accounts for 1/3 of TFP variation \citep{bloo17b}
        \item Firm-embedded productivity accounts for 28--34 percent of productivity differences \citep{burs09,alvi23}
    \end{itemize}

    \bigskip

    \main{Distributional accounting}
    \begin{itemize}
        \item Microdata used to quantify effects of expansion of education on evolution of income distribution
        \item Expansion dampened global inequality \citep{geth25}
    \end{itemize}

\end{frame}

\begin{frame}{Outline}

	\begin{enumerate}
	\item Overview of Basic Structure and Facts
	\item Measures of the Labor Input
	\item Measurement of Physical Capital
	\item The Constructive Approach to Human Capital
	\item Alternative Approaches to Human Capital
	\item Other Advances and Open Areas
    \item \main{Conclusion}
	\end{enumerate}

\end{frame}

\begin{frame}{Conclusion}

	\main{New measurement pushes inputs from 28\% $\rightarrow$ 45--65\% of GDP per worker differences}
	\begin{itemize}
	\item Hours deepen the puzzle (-9\%)
    \item Physical capital unchanged or slightly smaller (-5\% to 5\%)
    \item Human capital much larger (60--70\%); constructive and deductive approaches agree
	\end{itemize}

	\bigskip

	\main{Many open questions}
	\begin{itemize}
    \item Revisiting assumptions, improving existing measures
    \item Some important dimensions remain highly uncertain (public capital, imperfect substitution)
	\end{itemize}

	\bigskip

	\main{Emphasize limitations}
	\begin{itemize}
	\item Needs external evidence on input gaps and how to scale them
	\item Proximate, not root, causes; no direct policy levers
	\end{itemize}

\end{frame}

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\end{document}
