3.1 Model specifications

To calculate $R_{World}$ and $R_{c,US}$ , we need to calculate the hypothetical output per worker, $y_{c}^{H^{*}}$ , which differs from the actual output per worker, $y_{c}^{actual}$ , only in that human capital is reallocated across sectors so that gaps in the marginal product values are eliminated.Footnote ⁸ In each country, aggregate output per worker, $y_{c}$ , is defined as

(4) \begin{equation} y_{c}=\frac {Y_{c}}{N_{c}} \end{equation}

where $N$ is the total number of workers. We assume that there is a single final good $Y$ produced by a representative firm in a perfectly competitive and undistorted final output market. This firm combines the value added $Y_{i}$ of $i=1,\ldots \,,J$ sectors using a Cobb–Douglas production technologyFootnote ⁹

(5) \begin{equation} Y_{c}=\prod _{i=1}^{J}Y_{i}^{\upsilon _{i}}, \quad\text{where}\sum _{i=1}^{J}\upsilon _{i}=1 \end{equation}

In each sector, a representative firm produces value added using a constant-returns-to-scale CES production function of the form (the country index is omitted for clarity)

(6) \begin{equation} Y_{i}=C_{i}\left [\beta _{i}H_{i}^{\rho _{i}}+(1-\beta _{i})K_{i}^{\rho _{i}}\right ]^{\frac {1}{\rho _{i}}} \quad {\beta _{\text{i}}}{{\in }}(0,1), \ \rho _i{{\in }}(\!-\!\infty , 1) \end{equation}

where $K$ is physical capital, $C$ is a (Hicks-neutral) efficiency parameter, $\beta$ is the distribution parameter and the elasticity of substitution between physical and human capital is defined as $\sigma =1/(1-\rho )\gt 0.$ Note that the production function (6) becomes Cobb–Douglas as a special case if $\sigma \rightarrow 1$ while $\sigma \lt 1$ implies that factors are gross complements in production. We assume a Hicks-neutral technology for simplicity but distribution parameters may implicitly capture factor-biased technology.

Furthermore, we define $H_{i}$ in each sector as a human capital input aggregator

(7) \begin{equation} H_{i}=F\left (LH_{i},LM_{i},LL_{i}; \ \gamma _{i}\right ) \end{equation}

that is defined by the hours worked of low-skilled ( $LL$ ), medium-skilled ( $LM$ ), and high-skilled ( $LH$ ) workers and the elasticity of substitution between the skill types defined as $\eta =1/(1-\gamma )\gt 0$ . Building on (7), we modify (6) to a two-level nested CES production function of the formFootnote ¹⁰

(8) \begin{multline} Y_{i}=C_{i} {\left [\beta _{i} \underbrace {\left [\alpha _{i}^{LH}\left (LH_{i}\right )^{\gamma _{i}}+\alpha _{i}^{LM}{\left (LM_{i}\right )}^{\gamma _{i}}+(1-\alpha _{i}^{LM}-\alpha _{i}^{LH}){\left (LL_{i}\right )}^{\gamma _{i}}\right ]^{^{\frac {\rho _{i}}{\gamma _{i}}}}}_{human \ capital \ nest \ for \ H_i^{\rho _i} \ in \ eq. \ (6)} + (1-\beta _{i})\left (K_{i}\right )^{\rho _{i}}\right ]}^{\frac {1}{\rho _{i}}} \end{multline}

Here, $\alpha ^{LM}$ , $\alpha ^{LH}$ and $\beta$ are the distribution parameters, while $\rho$ and $\gamma$ govern the substitutability between inputs. Each function is assumed to be “well-behaved” such that it satisfies standard regularity conditions like continuity, differentiability and is strictly increasing and concave in all factors. Note that in the traditional approach where $\gamma \rightarrow 1$ (i.e., perfect substitution at fixed ratios), the human capital input is calculated as low-skilled labor equivalents by weighting the hours worked by each skill type with its relative efficiency, captured by the distribution parameters $\alpha ^{LM}$ and $\alpha ^{LH}$ . Importantly, this means that the relative marginal productivity of each skill type is independent from the skill composition of the labor force. One contribution of this paper is to relax this assumption and work with the generalized function (8) to allow $1\lt \eta \leq \infty$ (i.e., imperfect substitutability). This facilitates a stronger connection of our misallocation results to the literature on human capital in development accounting (see references in footnotes 10 and 23) and provides an additional robustness check of our results.

3.2 Optimization problem

In each country $c$ , the optimal allocation of each skill type $L_{k}^{*}\,\in \,\left \{ LL,LM,LH\right \}$ across sectors, ceteris paribus, can be determined by solving the problem of a social planner who maximizes aggregate value added.Footnote ¹¹

(9) \begin{equation} L_{ki}^{*}=\underset {{\scriptscriptstyle L_{ki}}}{\arg \;\max }\;Y_{c} \end{equation}

The first order condition of this maximization problem is that the value of the marginal product (MPV) of each skill type $k$ is equated across sectors.Footnote ¹² In reality, mobility barriers distort the allocation of workers across sectors. The possible sources of these barriers are numerous, including sector-specific labor regulations, skill requirements, or direct mobility costs (non-monetary and monetary). These distortions cause a wedge between the current MPV in each sector and the hypothetical sector-spanning MPV obtained at an efficient allocation, so that

(10) \begin{equation} MPV_{i}^{L_{k}}=\big(1+\lambda _{i}^{L_{k}}\big)MPV{}_{hyp}^{L_{k}} \quad\forall i \end{equation}

where $\lambda _{i}^{L_{k}}$ displays the wedge. The setup presented in equation (10) is the starting point to find the hypothetical $y_{c}^{H^{*}}$ under allocative efficiency of human capital. Following Aoki (Reference Aoki2012), we use equation (10) and $L_{kc}=\sum _{i=1}^{J}L_{kci}$ , to write the current sector allocation of human capital within each country (omitting the country index c for clarity) as

(11) \begin{equation} L_{ki}=\frac {\frac {\left (1+\lambda _{i}^{L_{k}}\right )MPV_{hyp}^{^{L_{k}}}L_{ki}}{\left (1+\lambda _{i}^{L_{k}}\right )MPV_{hyp}^{^{L_{k}}}}}{\sum _{j=1}^{J}\left (\frac {\left (1+\lambda _{j}^{L_{k}}\right )MPV_{hyp}^{^{L_{k}}}L_{kj}}{\left (1+\lambda _{j}^{L_{k}}\right )MPV_{hyp}^{^{L_{k}}}}\right )}L_{k}, \end{equation}

which can be written as

(12) \begin{equation} L_{ki}=\frac {MPV_{i}^{L_{k}}L_{ki}}{\sum _{j=1}^{J}\left (MPV_{j}^{L_{k}}L_{kj}\right )}\tilde {\lambda }_{i}^{L_{k}}L_{k}, \end{equation}

where $\tilde {\lambda }_{i}^{L_{k}}$ does not display the absolute, but relative wedge for each sector. The optimal quantity of each skill type in each sector can be obtained by setting $\tilde {\lambda }_{i}^{L_{k}}$ equal to unity. That is, we assume that sector-specific distortions do not bias the allocation of skill types across sectors.

Some additional remarks are in order. A major assumption underlying the reallocation exercise is that human capital (or labor skill type respectively) is perfectly substitutable across sectors within each country. Clearly, this assumption is strong. Note, however, that if workers in each sector are seen as idiosyncratic, this would render an investigation of human capital misallocation obsolete. Further note that equating the MPV of human capital instead of raw hours worked or number of workers allows for the possibility that wages per hour or worker across sectors may vary because of cross-sector differences in actual human capital. In the case of skill types being imperfect substitutes, optimization within skill types allows for wage gaps between different skills types after reallocation.

3.3 Cobb–Douglas vs. CES

The optimal sector allocation of each factor is essentially determined by the sector’s relative value added and factor output elasticity. If one assumes that both consumer preferences and production in each sector are best represented by a Cobb–Douglas function, the optimal sector allocation can be readily obtained from equation (12). This is facilitated by the constant output elasticity of human capital in the Cobb–Douglas production function, derived as $\alpha = (\partial Y/\partial H) \cdot H/Y$ from equation (1). Labor’s marginal product value is then given as:

(13) \begin{equation} MPV^{L_k}_i = \alpha _i \times \frac {Y_i}{L_{ki}}. \end{equation}

Rearranging for labor and noting that the $MPV$ needs to be equal for all all sectors in an optimum without complete specialization shows that the optimal labor allocation will be proportional to $\alpha _i Y_i$ :

(14) \begin{equation} L_{ki}^* \propto \alpha _i Y_i. \end{equation}

Note that the right-hand side of the optimality condition (14) does not depend on labor.Footnote ¹³ This is no longer the case with the more general CES production function (6), where the output elasticity $(\partial Y/\partial H) \cdot (H/Y) = \beta \cdot \frac {H^\rho }{\beta H^\rho + (1-\beta )K^\rho }$ leads to:

(15) \begin{equation} MPV^{L_k}_i = \beta _{i} \frac {L_{ki}^{\rho }}{(1-\beta _i) K_i^{\rho } + \beta _i L_{ki}^{\rho }} \times \frac {Y_i}{L_{ki}}, \end{equation}

in which case labor $L$ can no longer be meaningfully isolated and the $MPV$ depends in a nontrivial way on $L$ , relative to $K$ and $\beta$ , and the parameter $\rho$ (unless $\rho \rightarrow 0$ , which is the special Cobb–Douglas case). This problem further complicates in the nested CES case of equation (8).

In our empirically relevant case of $\rho \lt 0 \ (\Leftrightarrow \sigma \lt 1)$ , the $MPV$ (15) of labor (and its associated output elasticity) declines in $L$ for fixed $K$ . Intuitively, this means that productivity gains from reallocating labor (and its human capital) decline faster in this CES case than under the Cobb–Douglas benchmark (even though the level of those gains may be initially higher due to factor complementarity).

Keeping capital $K$ fixed in each sector and exclusively focusing on human capital mis- and reallocation allows us to isolate the complex interactions implied by equation (15) and aligns with earlier seminal studies in the literature on human capital misallocation (Vollrath Reference Vollrath2014; Herrendorf and Schoellman Reference Herrendorf and Schoellman2018). Such a “short-run capital specificity” approach is also highly informative from a policy perspective if one believes that labor market distortions are the key reason for allocative inefficiency (e.g., ElFayoumi Reference ElFayoumi2022) since it adds to the understanding of adjustment mechanisms (see Neary Reference Neary1978a, Reference Nearyb). We discuss potential implications of capital mobility across sectors in subsection 5.2.

3.4 Optimization in the CES case

The preceding discussion emphasizes the restrictiveness of using the Cobb–Douglas production function as a very special CES case. But it also highlights that the higher flexibility of the CES framework requires some methodological adjustments. One challenge is to find the most plausible CES parameters, which we econometrically obtain through a normalized supply-side system of nonliear equations explained in the next section. The other challenge is that the optimal sector allocation of human capital can no longer be determined analytically if the output elasticity of human capital is not constant (as explained above). We therefore rely on an iterative numerical solution based on equation (12) through an algorithm explained in online appendix D.

To be more specific, we rely on a “normalized” version of the CES production function. This is beneficial in terms of econometric identification and for calibrating the distribution parameters through factor shares (see León-Ledesma et al. Reference León-Ledesma, McAdam and Willman2010). We hence replace the sector production functions in (6) and (8) by the normalized functions:

(16) \begin{equation} Y_{i}=Y_{0i}\left [\phi _{0i}\left (\frac {H_{i}}{H_{0i}}\right )^{\rho _{i}}+(1-\phi _{0i})\left (\frac {K_{i}}{K_{0i}}\right )^{\rho _{i}}\right ]^{\frac {1}{\rho _{i}}} \end{equation}

and

(17) \begin{align} Y_{i}=Y_{0i}\left [\phi _{0i}\Omega _{i}^{^{\frac {\rho _{i}}{\gamma _{i}}}}+(1-\phi _{0i})\left (\frac {K_{i}}{K_{0i}}\right )^{\rho _{i}}\right ]^{^{^{\frac {1}{\rho }_{i}}}} \\[-10pt] \nonumber \end{align}

where $\Omega _{i}=\left [\theta _{0i}^{LH}\left (\frac {LH_{i}}{LH_{0i}}\right )^{\gamma _{i}}+\theta _{0i}^{LM}\left (\frac {LM_{i}}{LM_{0i}}\right )^{\gamma _{i}}+\left (1-\theta _{0i}^{LH}-\theta _{0i}^{LM}\right )\left (\frac {LL_{i}}{LL_{0i}}\right )^{\gamma _{i}}\right ]$ . The subscript 0 stands for the point of normalization.

Given the production function (16), the optimal allocation of human capital input in each sector is then determined by (see online appendix C for derivation):

(18) \begin{equation} H_{i}^{\ast }=\frac {Y_{i} \phi _{0i}\left (\frac {H_{i}}{H_{0i}}\right )^{\rho _{i}}\left [\phi _{0i}\left (\frac {H_{i}}{H_{0i}}\right )^{\rho _{i}}+(1-\phi _{0i})\left (\frac {K_{i}}{K_{0i}}\right )^{\rho _{i}}\right ]^{-1}}{\sum _{j=1}^{J}\left (Y_{j} \phi _{0j}\left (\frac {H_{j}}{H_{0j}}\right )^{\rho _{j}}\left [\phi _{0j}\left (\frac {H_{j}}{H_{0j}}\right )^{\rho _{j}}+(1-\phi _{0j})\left (\frac {K_{j}}{K_{0j}}\right )^{\rho _{j}}\right ]^{-1}\right )} \ H. \end{equation}

Note that the numerator in equation (18) is sector output (value added) $Y_i$ followed by the output elasticity of human capital (the first derivative of our normalized CES production function (16) w.r.t. $H_i$ ). Because the output elasticity multiplied with $Y/H$ defines the MPV, this is the CES version of the numerator in equation (12). The same logic applies to the denominator (and recall that $\tilde {\lambda }\equiv 1$ in the optimum).

Similarly, in the nested CES case of sector production functions with the generalized human capital aggregate, the optimal allocation of each skill type is found by a normalized analog of equation (12):

(19) \begin{equation} L_{ki}^{\ast }=\frac {Y_{i} \phi _{0i}\theta _{0ki}\left (\frac {L_{ki}}{L_{0ki}}\right )^{\gamma _{i}}\Omega _{i}^{-1}\varGamma _{i}^{\frac {\rho _{i}-\gamma _{i}}{\gamma _{i}}}}{\sum _{j=1}^{J}\left (Y_{j} \phi _{0j}\theta _{0kj}\left (\frac {L_{kj}}{L_{0kj}}\right )^{\gamma _{j}}\Omega _{j}^{-1}\varGamma _{j}^{\frac {\rho _{j}-\gamma _{j}}{\gamma _{j}}}\right )}\ L_{k} \end{equation}

where $\Omega _{i}=\left [\phi _{0i}\varGamma _{i}^{^{\frac {\rho _{i}}{\gamma _{i}}}}+(1-\phi _{0i})K_{i}^{\rho }\right ]$

and $\varGamma _{i}=\left [\theta _{0i}^{LH}\left (\frac {LHi}{LH_{0i}}\right )^{\gamma i}+\theta _{0i}^{LM}\left (\frac {LMi}{LM_{0i}}\right )^{\gamma i}+\left (1-\theta _{0i}^{LH}-\theta _{0i}^{LM}\right )\left (\frac {LLi}{LL_{0i}}\right )^{\gamma i}\right ].$

We opt for sector-country-specific averages of $Y$ , $K$ , $H$ , $LH$ , $LM$ , and $LL$ as normalization points in our exercise. This rationale follows Klump et al. (Reference Klump, McAdam and Willman2007), who argue that appropriate values for the distribution parameter are best detected from the data and that averages also net out short-term fluctuations.Footnote ¹⁴

4.1 Output and input data

All output and input data used in this study come from the Socio-Economic Accounts in the World-Input-Output Database (WIOD-SEA).Footnote ¹⁵ The WIOD-SEA database has been constructed on the basis of national accounts data and harmonization procedures were applied in order to ensure international comparability. For the period 1995–2007, WIOD-SEA contains annual data for 35 industries in 40 countries (27 EU countries and 13 other major countries) on gross output and intermediate inputs at current basic prices, capital stock, as well as hours worked and labor compensation by skill type (low-, medium-, and high-skilled) (Timmer et al. Reference Timmer, Dietzenbacher, Los, Stehrer and Vries2015). Since information on the latter is not available at the industry level for a large number of countries, we aggregate the 35 industries into 14 sectors. Moreover, we drop the Mining and Quarrying sector as well as Public Utilities sector because they are unlikely to absorb large quantities of workers in reality, irrespective of their (marginal) labor productivities. The motivation to do so is that both sectors have high labor productivity levels, but are unlikely to absorb large quantities of workers in reality. This would considerably inflate the estimated (hypothetical) reallocation gains. The final sample therefore consists of the 12 sectors listed in Table 1.

Table 1.

Sector overview

Table 1. Long description

Table with columns for abbreviations, sector names, and ISIC Rev 3.1 descriptions. The abbreviations include AtB, D, F, G, H, I, J, K, L, M, N, and OtP. The sectors listed are Agriculture, Manufacturing, Construction, Trade Service, Hotels & Restaurants, Transport & Storage, Finance Services, Business Services, Public & Social Services, Education, Health & Social Work, and Other Services. Each sector has a corresponding description based on ISIC Rev 3.1.

A central issue for cross-country comparison of output and productivity levels is that output is expressed in international purchasing power parities (PPP). To account for the substantial variation in relative prices across industries, we use multilateral industry-level PPPs from Inklaar and Timmer (Reference Inklaar and Timmer2014) (see Table A.2 in the online appendix). As no information on relative prices are available for Taiwan, the sample reduces to 39 countries. Also, relative price data is only available for the year 2005. We therefore proceed as follows. We deflate both gross output and inputs using the industry-level price deflators integrated in WIOD-SEA and obtain growth rates of both series. An advantage of the double-deflated value-added method, compared to a single-deflated value-added measure, is that it can account for changes in the relative prices of intermediate inputs to gross outputs. The real growth rates are then used to extrapolate the gross output and intermediate input series, which are converted into PPPs for the available year 2005.Footnote ¹⁶ In a last step, we construct a value-added series in constant 2005 PPPs by subtracting the value of intermediate inputs from gross output, both in constant PPPs. For the physical capital stock, we use growth rates from the real capital stock data measured in national currencies in WIOD-SEA and the exchange rate from PWT 8.1 (Feenstra et al. Reference Feenstra, Inklaar and Timmer2015) to convert the series into constant 1995 US$.

The data on hours worked in WIOD-SEA includes both self-employed and employees.Footnote ¹⁷ Worker skill types are classified on the basis of educational attainment levels as defined in the International Standard Classification of Education (ISCED): low-skilled (ISCED categories 1 and 2), medium-skilled (ISCED 3 and 4), and high-skilled (ISCED 5 and 6). We define the general normalized human capital input in country $c$ in sector $i$ as L

(20) \begin{equation} \left (\frac {H_{ci}}{H_{0ci}}\right )^{{{\gamma }}_{ci}}=\left [\theta _{0ci}^{LL}\left (\frac {LL_{ci}}{LL_{0ci}}\right )^{{{\gamma }}_{{ci}}}+\theta _{0ci}^{LM}\left (\frac {LM_{ci}}{LM_{0ci}}\right )^{{{\gamma }}_{{{ci}}}}+\theta _{0ci}^{LH}\left (\frac {LH_{ci}}{LH_{0ci}}\right )^{{{\gamma }}_{{{ci}}}}\right ]^{\frac {1}{{\gamma{ci}}}} \end{equation}

where the subscript $0$ indicates the normalization point defined as the average over the period 1995–2007. Recall that both the Cobb–Douglas and the standard two-factor CES production function implicitly assume $\gamma =1$ (i.e., perfect substitutability of skill types at fixed ratios), in which case equation (20) can be rewritten as

(21) \begin{equation} \frac {H_{ci}}{H_{0ci}}=\theta _{0ci}^{LL}\times \left [\frac {LL_{ci}}{LL_{0ci}}+\frac {\theta _{0ci}^{LM}}{\theta _{0ci}^{LL}}\frac {LM_{ci}}{LM_{0ci}}+\frac {\theta _{0ci}^{LH}}{\theta _{0ci}^{LL}}\frac {LH_{ci}}{LH_{0ci}}\right ] \end{equation}

Further, note that the general normalized form in (21) includes the implicitly normalized human capital input $H_{ci}$ if $H_{0ci}=LL_{0ci}=LM_{0ci}=LH_{0ci}=1$ . In this case, $\theta _{0ci}^{LL}$ , $\theta _{0ci}^{LM}$ , and $\theta _{0ci}^{LH}$ collapse to measures of relative wages.Footnote ¹⁸ The general normalized human capital input thus incorporates the standard human capital input measured as unskilled worker equivalents with different worker types being weighted by their relative wage. Importantly, however, $\frac {H_{ci}}{H_{0ci}}$ also contains $\theta _{0ci}^{LL}$ , which measures the contribution of low-skilled workers to human capital services in sector $i$ in country $c$ .Footnote ¹⁹ In growth or development accounting exercises, $\theta _{0ci}^{LL}$ is often (implicitly) assumed to be fixed across time or equal across countries and can therefore be ignored (see, e.g., Jones Reference Jones2014).Footnote ²⁰ Conversely, by working with the nested CES function in equation (19) directly, we implicitly assume that $\theta _{0ci}^{LL}$ differs across sectors and countries. Considering $\theta _{0ci}^{LL}$ in equation (21) is thus important to ensure consistency, and therefore, allow a direct comparison between the obtained cost of misallocation based on the different specifications.

4.2 Distribution parameters

Aside from output and input data, we need values for the distribution parameters. Since WIOD-SEA contains information on labor compensation at the sector level, we calibrate $\phi _{0ci}$ as the geometric average labor share in value added (measured in current prices in national currency) in sector $i$ in country $c$ over all time periods $T$ . This approach is common in the context of normalized CES production functions. Notably, WIOD-SEA also provides information on the compensation shares for each skill type. This allows us to further calculate $\theta _{0ci}^{LM}$ , and $\theta _{0ci}^{LH}$ based on the relative skill-shares in total labor compensation of each skill type working in sector $i$ in country $c$ at the normalization point. Specifically, we calculate

(22) \begin{align} \theta _{0ci}^{LH}=\left [\frac {\bar {w}_{ci}^{LH}\bar {LH_{ci}}}{\bar {w}_{ci}^{LL}\bar {LL_{ci}}+\bar {w}_{ci}^{LM}\bar {LM_{ci}}+\bar {w}_{ci}^{LH}\bar {LH_{ci}}}\right ] \\[-8pt] \nonumber \end{align}

(23) \begin{align} \theta _{0ci}^{LM}=\left [\frac {\bar {w}_{ci}^{LM}\bar {LM_{ci}}}{\bar {w}_{ci}^{LL}\bar {LL_{ci}}+\bar {w}_{ci}^{LM}\bar {LM_{ci}}+\bar {w}_{ci}^{LH}\bar {LH_{ci}}}\right ] \\[-8pt] \nonumber \end{align}

(24) \begin{align} \theta _{0ci}^{LL} =1-\theta _{0ci}^{LM}-\theta _{0ci}^{LH} \qquad\qquad \qquad \\[12pt] \nonumber \end{align}

where the bars indicate geometric averages over time.

Despite the effort put into the construction of WIOD-SEA, important limitations exist and warrant discussion. It is a well-known problem that hours worked and labor income of self-employed is not registered in the National Accounts in many countries. As discussed in detail in online appendix B, WIOD-SEA addresses this issue through imputation. A drawback of this imputation is that it can result in economically unreasonable labor revenue shares. Indeed, in 16 out of the 486 country-sector observations, $\phi _{0ci}$ exceeds the value of 1. We deal with these questionable parameter values in the baseline counterfactual by setting the maximum value of $\phi _{0ci}$ to 0.9. Admittedly, this ad hoc adjustment is somewhat arbitrary. We therefore run robustness tests, where we treat $\phi _{0ci}$ as a structural parameter that is common across countries.Footnote ²¹ It is worth noting that the relative efficiency parameters of skill types, $\theta _{0ci}^{LM}$ and $\theta _{0ci}^{LH}$ , are unaffected by a potential bias in $\phi _{0ci}$ as long as there is no systematic relation between compensation shares and self-employment.Footnote ²²

4.3 Elasticity of substitution parameters

Lastly, we require values for the unobserved substitution parameters, $\sigma$ and $\eta$ . While there is now a large set of estimates for the aggregate elasticity of substitution between physical and human capital, evidence for the sector level is scarce. Also, the few available estimates do not directly map to the sectors considered in this paper. A secondary contribution of this paper is thus to estimate elasticities of substitution on the sector level that can then be used in the reallocation exercise. More precisely, we identify the parameters based on a normalized supply-side system of equations (Klump et al. Reference Klump, McAdam and Willman2007; Herrendorf et al. Reference Herrendorf, Herrington and Valentinyi2015; McAdam and Willman Reference McAdam and Willman2018). As shown by León-Ledesma et al. (Reference León-Ledesma, McAdam and Willman2010), this approach is superior to single-equation estimation or factor-demand-equation systems, as it not only increases the degrees of freedom but also considers both technology (via the underlying production function) and optimization behavior (via FOC equations).

Despite the advantages of the normalized system approach, identifying two elasticities simultaneously in a highly nonlinear function remains challenging. The identification is further constrained by the fact that the harmonized sector data in WIOD-SEA only cover a rather short time horizon. Against this background, we focus on the estimation of $\sigma$ , which we assume to differ across sectors, but not across countries and time. To consider the impact of $\eta$ on $\sigma$ , we follow a grid search approach as proposed by Henningsen and Henningsen (Reference Henningsen and Henningsen2012). That is, we impose plausible estimates of $\eta$ that are motivated by the literature and obtain corresponding estimates for $\sigma$ .Footnote ²³ While clearly restrictive, note that these assumptions are implicitly also in use in all misallocation studies based on Cobb–Douglas, as this function implies $\sigma =1$ and $\eta \rightarrow \infty$ (see, e.g., Vollrath Reference Vollrath2009; Hsieh and Klenow Reference Hsieh and Klenow2009; Inklaar et al. Reference Inklaar, Lashitew and Timmer2017). Nonetheless, as a sensitivity test, we follow a more agnostic approach with regard to the true values of $\sigma _{ci}$ and $\eta _{ci}$ and compute bootstrap confidence intervals for $R_{c,USA}$ and $R_{World}$ .Footnote ²⁴

Figure 2.

Estimates of $\sigma$ .

For matter of space, the estimation specification of the normalized supply-side system and additional results are placed in online appendix E. The approach follows Trenczek and Wacker (Reference Trenczek and Wacker2026). Table 2 presents sector-specific estimates of $\sigma _{i}$ and corresponding 90%-confidence intervals (CI) based on bootstrap standard errors. Figure 2 additionally plots these estimates for $\eta \rightarrow \infty$ . Three findings stand out. First, in all sectors, and regardless of the imposed value of $\eta _{i}$ , $\sigma _{i}$ is estimated to be statistically significant below unity. Second, $\sigma _{i}$ is lower for lower values of $\eta _{i}$ and approximates the estimated value of the 1-level CES specification for $\eta \rightarrow \infty$ . Third, $\sigma _{i}$ is estimated quite precisely.

Table 2.

Sector elasticities of substitution with 90%-CI in parentheses

Table 2. Long description

A table comparing sector elasticities of substitution with 90%-confidence intervals for different values of eta. The table has 12 rows and 6 columns. Column headers are 2-Level CES with eta values of 1.6, 2, 4, 100, and 1-Level CES (implicit eta of infinity). Row labels are sector names including Agriculture, Manufacturing, Construction, Trade, Hotels & restaurants, Transport & storage, Financial intermediation, Business services, Public & social services, Education, Health & social work, and Other services. Each cell contains a value with a 90%-confidence interval in parentheses. Notable trends include varying elasticities across sectors and eta values, with some sectors showing higher elasticities at higher eta values.

Notes: Estimates of $\sigma$ (i.e., the elasticity of substitution between physical capital and the labor input aggregate with three skill types) based on different imposed values of $\eta$ . Estimation applies a two-step feasible nonlinear seemingly unrelated regression on a normalized supply-side system. 90%-CI in parentheses. Standard errors are bootstrapped based on 500 iterations.

There is some evidence to which we can compare these estimates, which are exactly in line with Chirinko (Reference Chirinko2008), but most of this evidence is restricted to the USA. For instance, Young (Reference Young2013) finds that the elasticity of substitution between physical and human capital is less than unity for a large majority of US 2-digit SIC level industries. Aggregating industries into sectors based on value-added-share-weights, Young (Reference Young2013) obtains an elasticity of 0.57 in the manufacturing sector and of 0.66 in services. Using capital price series and applying a low-pass filter to better capture the long-run substitution elasticity, Chirinko and Mallick (Reference Chirinko and Mallick2017) find industry-level estimates for USA to be below unity with most estimates being in a range of 0.2–0.7. Similar estimates are also found by Dissou et al. (Reference Dissou, Karnizova and Sun2015) in ten Canadian manufacturing industries for the period 1962–1997 and by Oberfield and Raval (Reference Oberfield and Raval2021), who recover elasticities for two-digit SIC US industries from plant-based estimates. The latter estimate a weighted average plant-level elasticity of 0.52 in manufacturing. Taking substitution across plants and industries into account, they recover an aggregate elasticity of 0.75 for the manufacturing sector in 2007. For broad sectors, Herrendorf et al. (Reference Herrendorf, Herrington and Valentinyi2015) find that capital and labor are gross complements in US manufacturing and services. Conversely, they estimate an elasticity of above one in agriculture. This stands in contrast to Mundlak’s (Reference Mundlak2005) examination of the US agriculture sector, advocating an elasticity of equal or below 1.Footnote ²⁵ ? also finds that labor and traditional capital are gross complements in sector-level EU KLEMS data. Overall, the sector estimates for the elasticity of substitution between physical capital and the human capital aggregate obtained in this paper seem to be fairly consistent with previous findings. Moreover, the estimates strongly suggest substitution elasticities that are inconsistent with Cobb–Douglas and in favor of a more flexible CES specification. It should finally be emphasized that long-run elasticities are usually higher than short-run elasticities (León-Ledesma and Satchi Reference León-Ledesma and Satchi2018; Koh and Santaeulalia-Llopis Reference Koh and Santaeulalia-Llopis2022).

5.1 Imperfect skill substitution

We now relax the implicit assumption that labor skill types are perfect substitutes (at fixed ratios). That is, we now rely on the nested CES production function (8) instead of the simple CES production function (6) and on the associated optimal allocation defined in (19) instead of (18).

Table 4 summarizes the hypothetical gains for the nested CES specifications. Rows (1)–(3) still impose $\sigma _{i}$ to be unity, while rows (4)–(6) consider different values of $\eta _{i}$ and a simultaneous change in the corresponding $\sigma _{i}$ . Some changes in the distributions are visible. However, if we fix $\sigma _{i}$ , the reallocation gains change only marginally for different values of $\eta _{i}$ , with notable exceptions in the case of India and Romania (see online appendix Figure A.1). Overall, the results suggest that the substitutability of physical capital and the human capital aggregate has larger quantitative implications than relaxing the assumption of perfect substitution (at fixed ratios) across skill types. This aligns with the finding of Hendricks and Schoellman (Reference Hendricks and Schoellman2023) that imperfect substitution of skills has limited consequences for cross-country output differences.

Table 4.

Gains from reallocation (other specifications, 2007)

Table 4. Long description

A table comparing hypothetical output per worker gains in percentage points across different specifications. The table has 6 rows and 8 columns. Column headers are Percentile, Min, 10th, 25th, Median, Mean, 75th, 90th, Max. Row labels are (1) to (6). Row 1: Percentile, Min, 10th, 25th, Median, Mean, 75th, 90th, Max. Row 2: (1), 2.0, 3.2, 3.7, 6.0, 10.5, 9.5, 18.6, 92.6. Row 3: (2), 1.9, 3.1, 3.6, 5.6, 10.0, 9.0, 17.5, 85.0. Row 4: (3), 1.8, 3.1, 3.5, 5.6, 9.8, 8.9, 17.3, 82.0. Row 5: (4), 1.4, 2.4, 3.1, 4.2, 7.3, 7.9, 14.3, 55.6. Row 6: (5), 1.3, 2.3, 2.8, 3.9, 6.9, 7.4, 13.2, 51.8. Row 7: (6), 1.3, 2.3, 2.8, 3.8, 6.8, 7.3, 13.0, 50.5.

Note: Hypothetical output per worker gains in percentage points. Calculation based on different specifications of normalized sector production functions. Note that $\eta$ is assumed homogeneous across sectors and determines $\sigma$ in the 2-level nested CES function in rows (4)–(6). All calculations based on value added measured in PPPs for 12 sectors.

5.2 Comparison to prior studies and capital mobility

There is limited evidence to which we can compare our estimates, as studies differ in their focus regarding input factors (physical and/or human capital), aggregation level (sector, industry, or firm), or outcome measure (TFP or output). It is thus informative to note that absolute reallocation gains would increase in our CES framework if we also allow for mobility of physical capital across sectors. In this case, capital would follow labor to more productive sectors and thus slows down the decline in labor’s $MPV$ .Footnote ²⁸

The seminal study by Hsieh and Klenow (Reference Hsieh and Klenow2009) calculates that improving factor allocation across firms to the extent observed in the United States could raise manufacturing productivity by 30–50% in China and 40–60% in India. Our exercise suggests aggregate gains from human capital reallocation of 58% (27.1%) for China and 105% (66%) for India in the Cobb–Douglas (CES) case. Yet, our study differs by the level of analysis (sector vs. firms), coverage (all sectors vs. within manufacturing; year), and factor mobility (human capital vs. labor and capital). Provided that productivity differences across sectors are large, especially in lower-income countries, and our counterfactual achieves full allocative efficiency of human capital (not just US efficiency), we consider our magnitudes for those two countries highly plausible despite the fact that keeping physical capital fixed provides a short-run lower bound of reallocation gains.

Inklaar et al. (Reference Inklaar, Lashitew and Timmer2017) estimate that the average observed manufacturing productivity level in their sample of 52 medium- and low-income countries would rise from 23% to 37% of the US level if cross-country differences in both capital and labor misallocation would be eliminated. This implies a 61% increase, which seems a bit high compared to our resultsFootnote ²⁹ but not implausible in overall magnitude, considering that our study differs in the sample composition, level of analysis, factor mobility etc.

Closer to our study, Vollrath (Reference Vollrath2009) estimates that variation across countries in the degree of human capital misallocation between the agriculture and non-agriculture sector could account for 30% of the variation in income per capita in his sample of 42 countries. Considering that Vollrath only looks at efficiency losses that occur between agriculture and the remainder of the economy, this estimate appears to be large, but it does not necessarily contradict ours. For one, Vollrath’s sample consists of several economies in which the agriculture sector employs a large share of total human capital and where the marginal product gaps of human capital between agriculture and industry are large (see Table 2 in Vollrath Reference Vollrath2009). This suggests that human capital misallocation can possibly explain a substantial part of output per worker differences in his sample. Moreover, Vollrath is forced to work with an arguably cruder sectoral measure of human capital. These differences considered, Vollrath’s empirical estimates are not inconsistent with our findings.

6.1 Robustness

6.2 Country dynamics in misallocation over time

One interesting finding depicted in Figure 3 is the stability in $R_{World}$ over the years. Figure A.12 in the online appendix indicates that this hides country dynamics in misallocation over time. The figure plots the hypothetical gains from reallocation for six countries. What we see is a mixed picture. Two countries, namely the Republic of Korea and the Czech Republic, improved the allocation of human capital across sectors over the 12 year period. On the contrary, in countries like Belgium, China, Greece, and India, the aggregate output costs of human capital misallocation in 2007 exceed the 1995 level. In light of the widespread structural change that occurred in China and India, this finding seems puzzling. However, underlying these developments are different records of human capital reallocation across sectors and productivity growth within sectors. While a thorough investigation is beyond the scope of this paper, it is worth noting that an increase in the aggregate costs of misallocation does not imply that there has been no growth-enhancing reallocation of workers across sectors over the period. In fact, in both China and India the reallocation of human capital across sectors contributed positively to growth, albeit just marginally (see online appendix table A.3).Footnote ³³ The dominant contribution came from within sector productivity growth. In fact, in both countries, marginal productivity values of human capital in agriculture remained low, despite substantial reallocation. Conversely, marginal productivity values increased further in other sectors. Consequently, gaps in marginal products across sectors increased. Rapid productivity growth within highly productive sectors can thus increase allocative inefficiency.Footnote ³⁴ At the same time, however, it raises the potential for growth-enhancing human capital reallocation.