On the elasticity of substitution between labor and ICT and IP capital and traditional capital

Layer 1: Overview

This paper estimates the elasticities of substitution between labor, information and communication technology (ICT) and intellectual property (IP) capital, and traditional capital using a nested constant elasticity of substitution (CES) production function. The motivation is twofold: standard macroeconomic models aggregate all capital into a single input and thus miss potentially distinct substitution relationships, and competing estimates of the labor-capital elasticity of substitution diverge sharply — with some finding gross substitutability (Karabarbounis and Neiman 2013) and others gross complementarity (Glover and Short 2020) — leaving unexplained the observed decline in labor income share across advanced economies.

The data come from the 2023 release of the EU KLEMS database for nine Euro Area economies (Austria, Belgium, Finland, France, Germany, Italy, Netherlands, Portugal, and Spain) over 1996-2020 (with Germany ending in 2019 and Portugal starting in 2001). The nesting structure places an ICT-IP capital aggregate (itself a CES nest of ICT equipment and IP capital, which includes software, databases, patents, and R&D capital) together with labor in an inner nest, and that combined aggregate is then nested with traditional capital in an outer nest. The rationale for grouping ICT and IP capital is their joint and complementary use — computers and software — and the observation that roughly 25% of granted patents in the sample period are ICT-related. Estimation follows the normalized CES methodology of Grandville (1989), Klump, McAdam, and Willman (2007), and Leon-Ledesma, McAdam, and Willman (2010), which jointly estimates the logged and normalized production function together with its first-order conditions using feasible generalized nonlinear least squares, weighting by country-year employment shares and correcting for heteroscedasticity and serial correlation. This approach is preferred because normalization anchors the point elasticity at sample averages and Monte Carlo evidence shows it outperforms first-order-condition-only or translog alternatives, especially when identifying factor-augmenting technological change alongside substitution elasticities.

The main results (Table 4, column 1) are as follows. The elasticity of substitution between labor and traditional capital (ε1) is estimated at 0.745 (standard error 0.009), statistically significantly below 1, implying gross complementarity. The elasticity between labor and the ICT-IP aggregate (ε2) is 1.187 (0.010), significantly above 1, implying gross substitutability. The elasticity between ICT and IP capital themselves (ε3) is 0.961 (0.003), significantly below 1, implying gross complementarity within the ICT-IP nest. The ICT capital-augmenting technological change parameter (γ_ICT) is estimated at 0.725, several orders of magnitude larger than the labor-augmenting parameter (γ_L = 0.003), consistent with rapid technological progress in ICT. The IP capital-augmenting parameter (γ_IP) is negative (−0.111), and the traditional capital-augmenting parameter (γ_TK) is negative but statistically insignificant (−0.002). For the US, ε2 is substantially larger at 1.712 (0.133), with ε1 = 0.724 (0.024) and ε3 = 0.922 (0.017).

A counterfactual accounting exercise (fixing ICT and IP technological progress indexes and capital stocks at their 1996 levels) finds that absent these developments, labor income share would have slightly increased in European countries rather than declining, and would have declined by about 75% less in the US over the sample period. ICT accumulation and technological progress is the dominant driver of the fall: absent ICT changes alone, labor share would have risen significantly in Europe.

The paper also derives the implied aggregate labor-capital elasticity (εL,K) using Hicks’s formula applied to the nested production function. The imputed εL,K for European countries ranges from approximately 1.36 to 1.43 over 1996-2020, rising through 1996-2008 and declining afterward. The US imputed values are substantially higher, ranging from approximately 2.14 to 2.37. By contrast, when the author directly estimates a two-input CES function combining labor with aggregate capital, the estimated elasticity is significantly below 1 (approximately 0.988 for European countries in the constant-CES specification), far below the imputed values. This divergence demonstrates that production function specification is consequential for identifying the labor-capital elasticity, and that models treating all capital as a single input can generate downward-biased estimates of this parameter.

Layer 2: Deep Dive

What is the identification strategy, and what are the main threats to it?

The author jointly estimates a normalized CES production function and first-order conditions (capital return equations and the wage equation) using feasible generalized nonlinear least squares with multiple starting points, selecting results by log likelihood, AIC, BIC, and R-squared. Normalization anchors the elasticity as a point elasticity at geometric sample averages, which is theoretically motivated and improves finite-sample identification. Main threats include: (1) endogeneity of factor inputs — the system of equations is estimated jointly but without instrumental variables, relying on non-arbitrage conditions to close the model; (2) negative estimates for γ_IP and γ_TK, which the author acknowledges may capture markups or capital underutilization rather than true technical change (Jiang and Leon-Ledesma 2018 show that omitting markups can bias the sign of capital-augmenting technology); (3) the US results are sensitive to initial values for the estimation algorithm, possibly because of the small sample size (24 observations); and (4) the counterfactual exercise abstracts from equilibrium effects and free-factor supply adjustments.

What are the main mechanisms distinguishing the three capital types, and how are they distinguished empirically?

ICT capital (computers, communication devices, peripherals) and IP capital (software, databases, patents, R&D capital) are grouped in an inner nest on the grounds of their complementary joint use. Traditional capital (machinery, transport, construction and structures) forms the outer nest. This nesting allows the elasticity of substitution between labor and the ICT-IP aggregate (ε2 > 1, gross substitute) to differ from the elasticity between labor and traditional capital (ε1 < 1, gross complement), which the paper argues is consistent with the automation literature’s emphasis on ICT displacing routine tasks. The elasticity of substitution within the ICT-IP nest (ε3 < 1) reflects gross complementarity between ICT equipment and IP assets (one needs software to use computers). The empirical distinction comes from the separate first-order conditions for each capital type, which link each capital’s income share to its stock and price, allowing the three elasticities to be separately identified.

What heterogeneity is documented across countries or time?

The main estimates pool 9 European countries weighted by employment shares; the author does not report country-by-country elasticity estimates but does report country-level descriptive statistics (Table I in the Data Appendix). Time-series heterogeneity is addressed through the imputed aggregate elasticity εL,K, which rises from approximately 1.367 in 1996 to a peak around 1.388-1.426 near 2008 (varying across the sensitivity columns of Table 6) and then declines to approximately 1.369-1.411 by 2020. The US elasticities are systematically higher than the European ones (εL,K ranging approximately 2.14-2.37 for the US vs. 1.36-1.43 for Europe; ε2 = 1.712 for the US vs. 1.187 for Europe). The time-varying aggregate capital specification in Table 7 shows the estimated ε1 for European countries follows an inverted-U shape over the sample period, while the US estimate shows the contrary pattern (though the latter is imprecise due to the small sample).

What robustness checks are run?

The paper estimates two alternative CES nesting structures (equations 20 and 21, reported in columns 2 and 3 of Table 4) to assess sensitivity to the nesting assumption. In specification (20), labor and traditional capital are nested first and then combined with the ICT-IP aggregate, so the elasticity between labor and ICT-IP equals that between traditional capital and ICT-IP. In specification (21), the different capital types are nested first and then combined with labor. Both alternatives confirm that ICT and IP capital are gross substitutes for labor. The paper also estimates a two-input labor-aggregate capital function in three variants: constant CES, elasticity as a linear function of compensation shares and relative prices, and elasticity as a quadratic polynomial of time (Table 7). Results using US data from the EU KLEMS database are reported separately (column 4 of Table 4 and columns 8-9 of Table 6). The imputed εL,K is further verified using data counterparts of the compensation shares rather than model-predicted shares (column 7 of Table 6), yielding essentially identical results with higher variability.

How does this paper relate to and differ from closely related prior work?

Relative to Karabarbounis and Neiman (2013), this paper agrees that labor and aggregate capital are gross substitutes (imputed εL,K > 1) and that capital deepening drives the labor share decline, but attributes the mechanism specifically to ICT and IP capital accumulation rather than the fall in all capital prices. It contrasts with Glover and Short (2020), whose below-1 estimates the paper reconciles by showing that treating all capital as a single input biases the aggregate elasticity downward. Relative to Eden and Gaggl (2018, 2019), who use US data and find ICT (including software) substitutes for labor in first-order-condition-only estimates, this paper adds normalization and biased technical change parameters and uses European panel data, and also separates ICT equipment from IP/software. Relative to Koh, Santaeulalia-Llopis, and Zheng (2020), who perform an accounting exercise attributing the labor share decline to IP capital capitalization, this paper provides structural estimates of substitution elasticities and corroborates the IP capital importance. Relative to Aum and Shin (2024), who use Korean firm-level data and find software substitutes for labor while ICT equipment complements it, this paper uses a different nesting (ICT and IP grouped together) and European aggregate data, and finds the combined ICT-IP aggregate is a gross substitute for labor — consistent with Aum and Shin’s software result driving the within-nest finding. The normalization approach distinguishes the paper from Antras (2004) and earlier aggregate studies that estimate only first-order conditions (which can produce upward-biased elasticity estimates when biased technical change is omitted).

What does the paper find about the source of the labor share decline, and what are the scope conditions on this result?

The counterfactual exercise (Section 4.2, Panel B of Table 3) finds that absent ICT and IP capital technological progress and accumulation, labor income share would have slightly increased in European countries over 1996-2020 rather than falling. Absent ICT changes alone, labor share would have risen significantly in Europe. The ICT-driven decline is the dominant contributor. By contrast, absent IP capital trends, labor share would have fallen substantially more (suggesting IP capital compensation growth, when attributed to capital rather than labor, partially offsets the ICT effect on labor’s share but its own share rise is the proximate driver of labor share decline). For the US, absent ICT and IP developments, labor share decline would have been about 75% smaller. Scope conditions: this is a static accounting exercise holding free factors at initial values and abstracting from general equilibrium effects. The results apply to total industrial value added (not individual sectors) and to the nine Euro Area countries in the sample. The exercise assumes the estimated production function parameters are the correct structural parameters, and thus inherits any limitations of the identification strategy.

What is the implication for the measured aggregate labor-capital elasticity, and why does it differ from standard estimates?

When the paper estimates a two-input (labor, aggregate capital) CES function directly, the estimated aggregate elasticity is significantly below 1 and close to estimates from Herrendorf, Herrington, and Valentinyi (2015). When it instead imputes the aggregate elasticity from the nested-CES parameter estimates using Hicks’s formula, the imputed values exceed 1 and are much larger. The paper shows analytically that εL,K > ε2 when the relative capital cost of ICT compared to traditional capital (pKICTKICT / pTKTK) takes sufficiently low values, which is the case in the data. This divergence arises because the single-input capital specification conflates the high substitutability of labor with ICT-IP capital and the low substitutability with traditional capital, yielding a biased estimate that depends on the capital composition. The paper concludes that production function specification is consequential for identifying the aggregate labor-capital substitution elasticity.

What are the key data features that drive the results?

ICT investment prices fell at an average annual rate of -4.6% relative to value added prices over the sample, while IP and traditional capital investment prices changed by -0.3% and +0.1% per year, respectively. Real ICT capital stocks grew at 4.9% per year, versus 3.4% for IP capital and 1.6% for traditional capital. ICT and IP capital depreciate rapidly (20.1% and 24.1% per year) compared to traditional capital (3.6%). These patterns imply computed rates of return on ICT capital that were very high at the start of the sample (131% in 1996, largely reflecting the fall in ICT prices that year) and fell sharply to 24% by 2020. The average share of labor and ICT-IP compensation in value added is approximately 71%, with labor making up about 92% of that combined share. The ICT share within the ICT-IP nest is about 21%, meaning IP capital compensation is substantially larger than ICT capital compensation.

Key Concepts

Allen-Uzawa elasticity of substitution: A point elasticity measuring the percentage change in the ratio of two inputs in response to a percentage change in their price ratio, holding output and other input prices constant. In this paper, it is estimated as a structural parameter of the nested CES production function, normalized at sample geometric averages; values above 1 imply gross substitutability and values below 1 imply gross complementarity.

Normalized CES production function: A CES specification that is indexed to sample averages of output and inputs so that the elasticity of substitution is defined as a point elasticity at those averages. This normalization, following Grandville (1989) and Leon-Ledesma et al. (2010), facilitates identification of both elasticity parameters and factor-augmenting technological change parameters, avoiding the conflation that arises in unnormalized specifications.

Gross substitutes / gross complements: Two inputs are gross substitutes (elasticity of substitution > 1) if a fall in the relative price of one leads to a rise in the share of cost devoted to it, reducing the other input’s cost share. They are gross complements (elasticity < 1) if a fall in relative price instead reduces cost share. In this paper, labor and ICT-IP capital are gross substitutes; labor and traditional capital and ICT with IP capital are gross complements.

Traditional capital (TK): In this paper’s taxonomy, all non-ICT, non-IP capital: machinery, transport equipment, construction, and structures. It is the residual capital category and is defined as a gross complement of labor in the estimated nested CES structure.

Intellectual property (IP) capital: Capital comprising software, databases, patents (including R&D capital), and other forms of intellectual property as measured in the EU KLEMS database. IP capital is grouped with ICT equipment in an inner CES nest on the grounds of complementary use. Its compensation share rise is the proximate accounting factor in the labor share decline.

Factor-augmenting technological change: Hicks-neutral or biased technical progress that enters multiplicatively with a specific factor input in the production function (e.g., γ_ICT for ICT capital), scaling the effective quantity of that input. In this paper, the ICT-augmenting parameter is estimated to be very large and positive (0.725), reflecting rapid ICT productivity growth, while IP- and traditional-capital-augmenting parameters are negative, which the author suggests may partly reflect markups or underutilization rather than pure technology.

Imputed aggregate labor-capital elasticity: The elasticity of substitution between labor and total capital derived analytically from the nested CES parameters using Hicks’s formula, rather than estimated directly from a two-input specification. In this paper, the imputed value exceeds 1 for Europe (~1.36-1.43) and is substantially higher for the US (~2.14-2.37), contrasting with directly estimated values that are below 1, illustrating the sensitivity of this parameter to production function specification.