Intangible capital and productivity divergence – Mouel – 2024 – Review of Income and Wealth

1 Introduction

The well-documented slowdown in productivity growth in all advanced economies has sparked a vivid debate around the question of whether the engine of economic growth has stalled (Syverson, 2017). This slowdown is accompanied by a number of trends symptomatic of declining business dynamism and growing concentration (Akcigit and Ates, 2019 and 2021). For example, firm entry rates, job reallocation rates, and the share of young firms in the economy have declined (Decker et al., 2016a and 2016b), as has the contribution of entrants and of creative destruction to growth (Garcia-Macia et al., 2019). Moreover, market concentration has increased in favor of firms making high profits and charging high mark-ups (De Loecker et al., 2020), having a high share of fixed costs in total costs (De Ridder, 2023), or having a low labor share of value-added (Autor et al., 2020). Finally, productivity is increasingly diverging between a group of top-performing “frontier” firms and the rest of the firm population (Andrews et al., 2015; Bahar, 2018; Andrews et al., 2019).

The deployment of intangible assets, also called Knowledge-Based Capital (KBC)1, has recently been put forward as a leading explanation behind many of these trends (Aghion et al., 2019; De Ridder, 2023). Aggregate investment in these productivity-enhancing assets has increased at a steady rate. Corrado et al. (2016) report that the annual growth rate of intangible investment, over the 2000–2013 period, was 2.6 percent in the United States and 2.0 percent across 14 European countries, outpacing tangible investment. However, this increase in total intangible investment is characterized by a highly skewed distribution of investment across firms (Arrighetti et al., 2014; Haskel and Westlake, 2018). On the one hand, large firms account for the bulk of total investment and, on the other, the majority of firms invest only very little. De Ridder (2023) argues that this pattern is driven by differences in the ability of firms to make productive use of intangible assets. Schumpeterian growth models predict that these differences in intangible investment will entrench differences in productivity performance, thereby disincentivizing follower firms from competing with frontier firms, reducing overall innovative effort and ultimately lowering aggregate productivity growth (Aghion et al., 2019; De Ridder, 2023).

Starting with the seminal study of Corrado et al. (2005), an ever-growing number of analyses have quantified investment in intangible assets at the macro and industry levels, and analyzed their contribution to economic growth and aggregate productivity (Roth (2019) lists 29 studies that make use of country-level or sector-level data). Building on this foundation, a smaller number of studies have analyzed the link between intangible capital and productivity (both labor and total factor productivity [TFP]) at the micro level.2 Table A.2 provides a comprehensive overview of this literature. The micro literature is unanimous: intangible capital has a positive effect on both firm output and productivity.3 However, these studies report average, rather than firm-specific results. The heterogeneous effect of intangible capital on productivity remains poorly documented.

In addition, empirical evidence linking firm-level behavior to aggregate trends in productivity growth and divergence remains scarce. Andrews et al. (2015) and Andrews et al. (2019) link productivity dispersion to the much discussed slowdown in productivity growth. They show that “frontier” firms, defined as firms in the top 5 percent of the global TFP distribution, achieve substantial productivity growth, whereas the rest of firms do not. This over-performance of frontier firms leads to a growing gap in productivity between the two groups and increasing productivity dispersion overall. Andrews et al. (2019) further provide econometric evidence that the growing divergence between the two groups is a key feature of the productivity slowdown at the aggregate level. This pattern of divergence is also documented by Berlingieri et al. (2017) in OECD countries, by Faggio et al. (2010) for the United Kingdom, and by Cette et al. (2018) for France.

The role of intangible capital as a driver of this pattern has been discussed, but not directly tested, in the context of some of the other trends discussed earlier, notably the rise of “superstar” firms (Autor et al., 2020), growing market concentration (Berlingieri et al., 2017), or declining business dynamics (Decker et al., 2016a). In addition, several studies have addressed the heterogeneity of the effect of investment in intangibles, such as Arrighetti et al. (2014) or Di Ubaldo and Siedschlag (2021), splitting the sample along different dimensions, such as firm size, and finding sizable differences. However, the effects of intangible capital within subgroups are still assumed to be identical across firms, and the heterogeneity is not linked to other phenomena.

The main objective of the present study is to fill these gaps. Specifically, we analyze two related research questions. First, we test whether the effect of intangible capital on firm-level productivity is indeed heterogeneous across firms. In particular, we explore whether, within industries, firms for which the marginal effect of intangible capital on productivity is higher are indeed those firms with larger stocks of intangible assets. This contributes to the literature by verifying one of the main assumptions of the theoretical models linking intangibles to productivity divergence between frontier firms and the rest. Second, we test whether the use of intangible capital at the firm level is associated with productivity divergence within industries, and whether this divergence is associated with slow productivity growth. Thus, our results strengthen our understanding of the causes behind the productivity slowdown and declining business dynamism.

The analysis uses firm-level administrative data from 2003 through 2014 for Germany, covering 47 detailed industries in the manufacturing and services sectors. Our measure of intangible capital encompasses software, R&D, organizational capital, and intellectual property products (IPP), such as patents, licenses, or trademarks.

The first stage of the analysis consists in estimating the role of intangible capital on firm-level productivity, using the control function approach along the lines of Doraszelski and Jaumandreu (2013) and Ackerberg et al. (2015). We use a flexible functional form to model this relationship and recover firm-specific estimates of the elasticity of intangible capital on productivity. The results show that the effect of intangibles on productivity is positive across all industries. Moreover, they confirm that this effect increases with the size of a firm’s stock of intangible capital. Stark differences in firm-specific elasticities of intangible capital can be found between firms with the top 5 percent largest intangible capital stock and the other firms in the industry. In addition, the elasticity of intangible capital in the bottom quintile of the distribution of intangible capital is close to zero or negative in around two-thirds of all industries. These results confirm the important heterogeneity in the ability of firms to reap the benefits of their intangible investments, further highlighting the productivity disadvantage of laggard firms that fail to reach a minimum scale of intangible capital.

The second stage of the analysis relates these firm-level findings to aggregate productivity trends, by focusing on the divergence of productivity between frontier firms and the rest. We adapt the framework of Andrews et al. (2019) and define the set of frontier firms as those with the top 5 percent largest stocks of intangible capital, for each industry and year. We find that the productivity of firms on the frontier grew faster than that of other firms: the gap in productivity growth amounted to 3 percentage points in manufacturing and 4.5 percentage points in services over a 5-year period. Correlating the size of the productivity gap at the industry level with the elasticities of intangible capital obtained from the production function estimations further emphasizes the role of intangibles as a driver of productivity divergence. Industries where the gap in productivity growth was largest are those industries where the average effect of intangible capital was higher. These industries also show more pronounced differences in the elasticity of intangible capital between frontier firms and the rest. Finally, this study sheds light on whether the intangible-driven divergence is related to the productivity slowdown. We regress the growth rate of productivity over a 3-year period in each industry on the increase in the productivity gap. The industries where the productivity gap increased the most are those industries that had lower average productivity growth. Taken together, these findings confirm that intangible capital fosters the divergence of productivity and is one of the forces behind the slowdown of productivity.

The remainder of the paper is structured as follows. In Section 2, we present the conceptual framework underlying our analysis, and discuss in detail the mechanisms linking heterogeneity in intangible capital to productivity divergence and to aggregate productivity performance. Section 3 describes the data set and provides descriptive statistics of intangible capital. Section 4 presents our structural model and the associated estimation strategy used to recover firm-level productivity and the elasticity of intangible capital. In Section 5, we discuss the results of the structural estimation, focusing on the effect of intangible capital on firm-level productivity, and some robustness estimations. In Section 6, we report the industry dynamics of productivity growth, on and off the frontier, and analyze its relationship to intangible capital. Finally, Section 7 presents our concluding remarks.

2 Conceptual Framework

The present analysis makes use of the Schumpeterian growth literature (Aghion and Howitt, 1992; Klette and Kortum, 2004) for its conceptual framework. In particular, the work of De Ridder (2023, p. 1) provides a model in which the increasing use of intangible assets in the economy simultaneously explains the productivity slowdown, the decline in business dynamism, and the rise in market power. The main mechanism at play is the heterogeneous ability of firms to make a productive use of intangible investments, which creates and entrenches “best versus rest” dynamics. Our analysis focuses on empirically linking the use of intangible capital at the firm level with these “best versus rest” dynamics observed at the industry level.

In the traditional endogenous growth framework, aggregate productivity increases each time a firm successfully innovates. By inventing a superior version of an existing good, a firm can become its monopoly supplier until the next successful innovation. In this framework, making a quality improvement is synonymous with making a productivity improvement: because all firms are assumed to have the same cost structure, the innovator offers a better product than its competitors for the same price (Aghion and Howitt, 1992).4 Entrants and incumbents both have an incentive to invest in innovation; the former in the hope of winning markets, the latter to escape the threat of entry.

Intangible assets change this mechanism in a fundamental way. De Ridder (2023) highlights two key characteristics of this type of assets: they “are scalable and firms differ in the efficiency with which they deploy them” (De Ridder, 2023, p. 1). Scalability refers to the effect of intangible capital on the cost structure of firms. While they represent an important up-front fixed cost, these investments allow firms to have lower marginal costs. This disconnects quality from productivity and introduces a new trade-off between quality improvements and reductions in marginal costs. The decline in marginal costs allows firms to charge lower prices and undercut rivals, and to obtain higher profits, which stimulates their innovation effort. De Ridder (2023) argues that if the deployment of intangible capital were evenly spread across all firms, this would lead to higher aggregate growth.

However, firms differ in the efficiency with which they deploy intangibles. Crucially, De Ridder (2023) introduces a firm-specific parameter that determines by how much a firm’s marginal costs will decrease for a given expenditure on intangible capital. Firms draw this parameter from a known distribution at birth, before making investment decisions. Firms that have drawn a higher intangible efficiency parameter will have more incentives to invest in intangible capital. De Ridder (2023) predicts and documents the existence in France and the United States of intangible-“superstars,” a small set of firms with a large stock of intangible capital. Our results presented in Section 5 further confirm this. Our data show the same concentration of intangible investment as reported in the literature (Arrighetti et al., 2014). In addition, we document that the effect of intangible capital on productivity increases with the size of a firm’s stock of intangible capital, supporting the choice to introduce an intangible efficiency parameter in the model.

It is this inequality in intangible efficiency, and the resulting concentration of intangible investment, that is responsible for dampened competition, innovation, and aggregate productivity performance. When faced with an incumbent that can offer low prices for a given quality level, because they have a large stock of intangible capital, innovators with lower stocks of intangibles will need to achieve a much larger quality improvement to be able to gain access to the market. Firms that have drawn a low intangible efficiency will be unable to catch-up to firms that have drawn a high-intangible efficiency. They will thus invest less in intangible capital, lower their innovation efforts, and consequently achieve lower productivity growth. The low intangible investments of followers ultimately reduce the incentive for leaders to aggressively push for more innovation, thereby lowering overall growth in the long run.

Building on these theoretical findings, our paper analyses the following two research questions. First, we test whether the effect of intangible capital on productivity is heterogeneous across firms and, in particular, whether it correlates positively with the size of the intangible capital stock within industries. The answer to this question makes an important contribution to the aforementioned theoretical literature, by providing evidence for the key assumptions in the model described earlier. Second, we explore the extent to which the use of intangible capital at the firm level explains some of the industry-level variation in productivity divergence and growth. To the best of our knowledge, our study is among the first to relate productivity divergence at the industry level with the heterogeneous effect of intangible capital on productivity at the firm level. Figure 1 summarizes the main mechanism described earlier and shows how the present paper provides evidence in support of this mechanism.

3 Data

The analysis uses the firm-level data sets collected by the German Statistical Offices and used for the construction of the aggregated data for the System of National Accounts (SNA). To ensure the greatest possible coverage of the German economy, we combine the AFiD Panel of Manufacturing Firms with the AFiD Panel of Service Firms.5 The full data set covers about 50 detailed industries in manufacturing (C), transport and warehousing (H), information and communication services (J), business services (M), and administrative activities (N).6 After data cleaning, the data set contains about 1 million firm-year observations. Complete coverage is available for the period 2009–2014 in the manufacturing sector, and for the period 2003–2013 for the services sectors. The panels are unbalanced.

We observe firm-level records of standard production variables, including gross value-added, investments in physical capital,7 the number of employees, material and energy expenses, and total wage bill. The data encompass the following intangible asset categories: software, IPP, R&D, and organizational capital, the latter being obtained from the Linked Employer-Employee Dataset (LIAB) of the Institute for Employment Research and added to the AFiD data.8 The analysis uses the sum of the four assets as the main variable of interest, acknowledging the fact that the optimal bundle of assets might differ across industries and firms. Nominal values are deflated using yearly price deflators at the two-digit industry level provided by the statistical office.

While the official firm-level data are of high quality, the final data set is not free of issues. First, the definition of the intangible asset is not identical in manufacturing and services due to the differences in the underlying surveys. However, by estimating the model at the 2-digit industry level, it is ensured that the different measurements are not pooled in the same estimation. We further acknowledge this issue by presenting all the results separately for manufacturing and services. Second, adding data on organizational capital to AFiD data adds noise to the intangible variable. This can potentially bias the estimation results. To address this issue, the estimation is conducted without organizational capital as part of the robustness checks presented in Section 5.2.

Table 1 reports the mean, the standard deviation, and selected percentiles of the distribution of intangible investments and capital stocks by 1-digit industries.9 The $$$ {10}^{\mathrm{th}} $$$ and $$$ {90}^{\mathrm{th}} $$$ percentiles and the median of the intangible variables demonstrate that there is large within-industry heterogeneity in the use of intangibles. In all industries, firms in the bottom decile invest little in intangibles and consequently have very small stocks. In the services industries, the median values of intangible investments and stocks are also close to zero, while across the manufacturing industries, the median values of intangible investments and stocks are €140,000 and €320,000, respectively. The fact that the distribution of intangibles is highly right-skewed is further visible in the fact that the mean is higher than the $$$ {90}^{\mathrm{th}} $$$ percentile in all industries apart from business services.

Figure 2 illustrates the heterogeneity in the use of intangible capital across and within detailed two-digit industries. The 47 sectors of the analysis are divided into two groups according to the median stock of intangible capital. The 26 sectors with the lowest median stock are plotted in the left panel, and the 21 sectors with the highest median stocks are plotted in the right panel.10 The difference in the range of the $$$ x $$$-axis between the left and right panels highlights the significant differences in the magnitude of intangible capital stocks across sectors. For example, in the computer and electronics industry (C26) the median value of intangible capital stock is €1.52 million, while it amounts to only €120,000 in the manufacturing of wood, wood products, and cork (C16). Similar differences can also be found between service sectors.

Figure 2 also showcases the skewness of the distribution of intangible capital stock within industries. In all sectors, the bottom quartile of firms reports close to null intangible capital stocks and the median value is often not much higher. The length of the upper whisker of the boxplots reveals that in each industry, some firms operate with a substantially higher intangible capital stock than the rest. This becomes even clearer when looking at the firms on the intangible capital frontier.11 For example, the pharmaceutical, motor vehicle, and other transport equipment industries (C21, C29, and C30) stand out, as the firms on the frontier have stocks exceeding €100 million. Among the other high-tech manufacturing industries,12 firms on the intangible frontier have stocks exceeding €20 million. Firms on the intangible frontier in the air transport, broadcasting activities, and management consultancies (H51, J60, and M70) services sectors have intangible capital stocks above €20 million.

Taken together, these results confirm that intangible capital is not evenly distributed but is concentrated among certain firms. This is consistent with the evidence of Arrighetti et al. (2014) and the predictions of De Ridder (2023), who states that firms with a higher ability to make use of intangible capital will invest heavily in these assets. However, this will be inhibiting the ability of followers to catch-up, thereby discouraging them from investing in the first place. We expect these entrenched differences in intangible capital effort to be reflected in differences in productivity performance.

4 Methodology

The analysis is centered around firm-level productivity, both in its relationship to intangible capital within firms and in its aggregate evolution. As first emphasized by Marschak and Andrews Jr. (1944), recovering productivity from production data is hampered by an inherent endogeneity problem, as firms have information on their productivity level when they make production decisions. From the econometric perspective, this unobserved factor, which is positively correlated with inputs, especially flexible inputs, and output, will introduce bias in the estimations. To address this issue, we implement the control function approach pioneered by Olley and Pakes (1996) and further developed by Levinsohn and Petrin (2003) and Ackerberg et al. (2015). We specify a structural model of production, where the different inputs of production have different adjustment costs.

With regard to the channel through which intangibles affect firms, two different approaches are taken in the literature. In the first one, intangibles are modeled as an input in the production function, on par with labor and capital (See e.g., Marrocu et al., 2012; Verbič and Polanec, 2014; Takizawa, 2015; Piekkola, 2016; Crouzet and Eberly, 2018; Kaus et al., 2020; Di Ubaldo and Siedschlag, 2021; Hsiao et al., 2021). The estimated elasticity thus captures the increase in output as a result of an increase in intangible capital.13 We acknowledge that the capitalization of intangible assets is consistent with their use as production inputs, just like physical capital. However, such an approach is incompatible with the hypothesis that intangible capital affects TFP, because using intangible capital as regular input requires the explicit assumption that it is uncorrelated with unobserved TFP, which is part of the error term in any firm-level estimation. In other words, once intangible capital is included as a normal input in a production function, any analysis of its impact on TFP is by definition inconsistent, a fortiori, any analysis of how its impact differs across firms.

Therefore, a second approach in the literature assumes that intangible capital directly affects the efficiency of production, i.e., TFP, rather than being a direct input in production (Crass and Peters, 2014; Ilmakunnas and Piekkola, 2014; Battisti et al., 2015; Mohnen et al., 2019). These studies draw on a branch of the innovation literature that assumes that R&D (one element of intangible assets) drives productivity (see,inter alia, Doraszelski and Jaumandreu, 2013).14 We follow this second approach and assume that intangible capital governs the evolution of productivity. As will be shown subsequently, we choose a specification that allows for firm-specific effects of intangibles. The main specification makes use of capitalized intangible assets, because data on intangible investments are more irregular, with more zero observations, and an exaggerated variation in the variable. However, we test for the sensitivity of this assumption by performing the analysis using investments rather than stocks as part of the robustness checks.

5 Effect of Intangible Capital on Firm-Level Productivity

5.1 Main Estimation Results

The results on the relationship between a firm’s stock of intangible assets and its productivity, obtained by estimating the law of motion of productivity, are presented in Table 2.20 The second and seventh columns contain the average values of $$$ \overline{\theta_c} $$$ by industry, describing the average percentage change in productivity from a 1 percent increase in the firm’s stock of intangible capital. The third and eighth columns contain the F-statistic for the test of joint significance of the three point estimates relating to intangible capital used in the calculation of $$$ {\theta}_c $$$, namely $$$ {\gamma}_c $$$, $$$ {\gamma}_{c^2} $$$, and $$$ {\gamma}_{c\omega} $$$.21 Together with the standard errors reported in parentheses, this gives an indication of the statistical significance of the calculated $$$ {\theta}_c $$$.

TABLE 2.
Average Marginal Effect of Intangible Capital on Productivity and Difference Between the Average Effect in the Top and the Bottom Quintiles, by Two-Digit Industry

Ind	$$$ {\overline{\hat{\theta}}}_c $$$	F stat	$$$ {\overline{\theta}}_c^{Q_T}-{\overline{\theta}}_c^{Q_B} $$$	N	Ind	$$$ {\overline{\hat{\theta}}}_c $$$	F stat	$$$ {\overline{\theta}}_c^{Q_T}-{\overline{\theta}}_c^{Q_B} $$$	N
C10	0.0048	34.79***	0.0083	9482	H49	0.0078	729.34***	0.0025	75957
	(0.00349)					(0.00155)
C11	$$$ – $$$.0036	2.66**	$$$ – $$$0.0039	1297	H50	0.0045	31.96***	0.0163	7454
	(0.00262)					(0.00766)
C13	0.0047	2.99**	0.0059	1929	H51	0.0169	39.76***	0.038	1482
	(0.00261)					(0.0159)
C14	0.0001	4.92***	$$$ – $$$0.0107	1053	H52	0.022	604.95***	0.0322	32179
	(0.00441)					(0.01282)
C15	0.0047	2.15*	0.0085	469	H53	0.0071	45.97***	0.0052	7820
	(0.00364)					(0.00315)
C16	$$$ – $$$0.0004	0.5	0.0009	1994	J58	0.0126	55.36***	0.0196	10931
	(0.0004)					(0.0086)
C17	0.0173	29.24***	0.0042	2257	J59	0.0218	63.9***	0.0326	5869
	(0.00632)					(0.0128)
C18	0.0117	17.81***	0.0205	1635	J60	0.0384	12.39***	0.0306	838
	(0.0085)					(0.01503)
C20	0.0117	35.98***	0.0036	4222	J61	0.0251	49.47***	0.0172	4212
	(0.00346)					(0.00764)
C21	0.0076	4.32***	$$$ – $$$0.0009	913	J62	0.0292	562.72***	$$$ – $$$0.018	43097
	(0.00342)					(0.00795)
C22	0.0057	23.09***	0.0075	4573	J63	0.0183	133.33***	0.0398	10904
	(0.00307)					(0.01781)
C23	0.0094	18.66***	0.0107	3763	M69	0.0067	438.28***	$$$ – $$$0.0044	81249
	(0.00486)					(0.00214)
C24	0.004	12.01***	0.0044	3275	M70	0.1053	1655.11***	0.0709	32536
	(0.00205)					(0.03616)
C25	0.0043	46.73***	0.0086	10430	M71	0.1126	3824.87***	0.0671	59139
	(0.0034)					(0.03481)
C26	0.0161	46.52***	0.007	3664	M72	0.0243	104.43***	0.0084	7208
	(0.00563)					(0.00891)
C27	0.0083	32.13***	0.0031	5063	M73	0.0409	638.7***	0.0677	26002
	(0.00279)					(0.0269)
C28	0.0071	77.09***	0.004	11500	M74	0.012	334.88***	0.0164	17242
	(0.00174)					(0.00755)
C29	0.0105	25.84***	0.0076	2974	M75	0.0058	41.98***	$$$ – $$$0.0026	11214
	(0.00466)					(0.00342)
C30	0.0018	1.13	0.0006	1000	N77	0.0147	228.95***	0.0288	18494
	(0.0009)					(0.013)
C31	0.0162	21.63***	$$$ – $$$0.0088	1875	N78	0.0459	485.6***	0.0974	16880
	(0.0051)					(0.03903)
C32	0.0064	19.66***	0.0126	3042	N79	0.0522	548.89***	0.069	14240
	(0.00505)					(0.02977)
C33	0.0043	10.44***	0.0088	3171	N80	0.1035	294.93***	0.0919	6746
	(0.00384)					(0.04343)
					N81	0.0122	765.15***	0.0498	48348
						(0.02188)
					N82	0.025	380.14***	0.0423	22407
						(0.01711)
					95	0.0096	123.06***	0.0511	8586
						(0.02191)

• Notes: Standard errors in parentheses. $$$ F $$$ statistic for the test of joint significance of the three coefficients: $$$ {\gamma}_c $$$, $$$ {\gamma}_{c^2} $$$, and $$$ {\gamma}_{c\omega} $$$. Names of industries can be found in Table A.1 in the Online Appendix.
• ***$$$ p<0.01 $$$, **$$$ p<0.05 $$$, *$$$ p<0.1 $$$.

In accordance with findings in the literature, the average effect of intangible capital on productivity is positive and significant across the economy: a larger stock of intangible capital is associated with higher productivity in the majority of industries.22 This confirms that the importance of intangible capital is not restricted to a few high-tech sectors in services and manufacturing.

In line with expectations, we find evidence of heterogeneity in the effect of intangible capital, both across and within industries. 201 for example, in the manufacturing sector, the average elasticity of intangible capital on productivity ranges from 0.016 in the computer and electronics industry (C26) to 0.005 in the leather product industry (C15). Likewise, large differences can also be found in the services sector. The elasticity of intangible capital ranges from 0.113 for architectural and engineering services (M71) to 0.005 in veterinary services (M75).

Moreover, this heterogeneity across industries in the effect of intangible capital is associated with the size of the intangible stock. We find a positive correlation between average $$$ {\theta}_c $$$ and the median intangible stock of the industry of 0.42 in the manufacturing sector and 0.74 in the services sector.

Speaking directly to our first research question, we also find strong evidence of heterogeneity in $$$ {\theta}_c $$$ across firms within industries and a positive correlation with intangible capital. To illustrate this, for each industry, we split the distribution of intangible capital stock into quintiles, as well as identify the group of firms with the top 5 percent largest stocks. For each of these six groups,23 we calculate the average $$$ {\theta}_c $$$. In the fourth and ninth columns of Table 2, we show the difference between the average elasticity of intangible capital in the top quintile $$$ {\overline{\theta}}_c^{Q_T} $$$ and the average elasticity of intangible capital in the bottom quintile $$$ {\overline{\theta}}_c^{Q_B} $$$. This difference is positive in 40 of the 47 industries, suggesting that within the same industry, the effect of intangible capital is larger, sometimes importantly so, in firms with larger stocks of intangible capital.

Figure 3 offers a visualization of these findings. Three patterns emerge, confirming the presence of important heterogeneity in the effect of intangible capital across firms.

TABLE 3.
Summary of the Four Types of Robustness Estimations

Functional form assumptions	Robust spec. (1)	Linear law of motion of productivity
	Robust spec. (2)	Translog production function
Data considerations	Robust spec. (3)	Omitting organizational capital
	Robust spec. (4)	Investment in intangible capital

First, as noted earlier, the effect of intangible capital on productivity increases with the size of a firm’s intangible capital stock. For example, in the computer and electronics industry (C26), a 1 percent increase in intangible capital increases productivity by 0.012 percent for firms in the bottom quintile of the intangible capital distribution, and by 0.019 percent for firms in top quintile. The elasticity of intangible capital is thus around 50 percent higher for firms at the top of the distribution compared to firms at the bottom. The increasing marginal effect is a general pattern in the majority of sectors.

Second, within the top quintile, frontier firms stand out as having an even higher elasticity of intangible capital. For example, in the computer and electronics industry (C26), the elasticity of intangible capital for frontier firms is 0.021, which is 12 percent higher than that of the other firms in the top quintile. The percentage difference in $$$ {\theta}_c $$$ between frontier firms and the other firms in the top quintile reaches 37 percent in the manufacturing of transport equipment (C30) and 40 percent in information services (J63). The difference between $$$ {\theta}_c $$$ on the frontier and at the median is even more marked. In the manufacturing sector, it ranges from 20 percent in the chemical industry (C20) to 75 percent in metal products (C25). In the services sector, the effect of intangible capital for frontier firms is at least double that for the median firm in seven industries and 30 percent or more for another 15 sectors.

Third, the marginal effect of intangible capital is negative in the bottom quintile of the intangible capital distribution in 14 industries and close to zero in another 15 of the 47 industries. Across manufacturing and services, the 14 industries with a negative $$$ {\theta}_c $$$ in the bottom quintile are those that report a stock of zero in the bottom decile in Table A.3.24 This pattern implies that there is a minimum scale for intangible capital to be effective, and that the lack of investment of certain firms can be due to their inability to obtain productivity benefits.

Overall, this empirical evidence answers our first research question and confirms the presence of large heterogeneity in the effect of intangible capital across firms and of increasing returns in the size of the intangible stock. It is in line with the assumptions of the theoretical models of De Ridder (2023) and Aghion et al. (2019), and supports the view that intangible capital is driving divergence between leaders and followers, in both their intangible efforts and productivity performance.

5.2 Robustness Estimations

We perform four robustness checks to test the sensitivity of our results with respect to two functional form assumptions and two choices regarding the data.25 As summarized in Table 3, we will refer to these four tests as “robust spec. (1)–(4).” For each of these four tests, we will compare results along two dimensions. First, whether we can replicate the generally positive effect of intangible capital on productivity from the main specification. These results are summarized in Tables A.6 and A.7. Second, whether we can replicate the heterogeneity in the elasticity of intangible capital within industries, especially the difference between $$$ {\overline{\theta}}_c^{Q_T} $$$ and $$$ {\overline{\theta}}_c^{Q_B} $$$. These results are summarized in Figure 4.

The Average Effect of Intangible Capital on Productivity

Robustness Check (1): Linear Law of Motion

The choice of a nonlinear functional form for the law of motion is driven by the research question derived in Section 2. To provide a benchmark for the average $$$ {\theta}_c $$$ reported in Table 2, the law of motion is also estimated using a linear functional form.26 The results of this first robustness check confirm the positive effect of intangible capital on TFP observed in the main specification, although the elasticity is generally of a smaller magnitude than the average $$$ {\theta}_c $$$ in Table 2.27

Robustness Check (2): Translog Production Function

This second robustness check addresses the sensitivity of the results to the assumed functional form of the production function. The combination of the flexible nonlinear specification for the law of motion of productivity with the restrictive Cobb–Douglas specification on the production function might create an issue. The variation stemming from the production function could appear in the law of motion, hence biasing the estimates for elasticity of intangibles. We therefore repeat the estimation using a translog production function instead of the Cobb–Douglas production function, keeping the nonlinear specification of the law of motion. The average $$$ {\theta}_{c, TL} $$$ reported in Tables A.6 and A.7 are nearly identical to the average $$$ {\theta}_c $$$ presented in Table 2. This suggests that the use of the Cobb–Douglas specification for the production function has no sizable impact on the estimated effect of intangible capital on TFP.

Robustness Check (3): Omitting Organizational Capital

The third robustness check tests the sensitivity of our results to our measure of organizational capital, by constructing intangible capital without it. The results confirm the positive and significant effect of intangible capital on productivity established in Section 5.1, thus providing reassurance that this positive association is not artificially driven by our measure of organizational capital. Nevertheless, the average elasticity of intangible capital without organizational capital, $$$ {\theta}_{c, noOC} $$$, is lower in magnitude in most industries (see the fifth and sixth columns of Tables A.6 and A.7).

There are two potential explanations for this finding. On the one hand, organizational capital is shown to positively affect firm performance (Bloom et al., 2013; Crass and Peters, 2014) and, therefore, its omission will lead to a lower estimated effect of total intangible capital. On the other hand, as the measure of organizational capital contains measurement error, part of the additional cross-industry variation in the intangible capital coefficient can be attributed to the larger noise in the variable. As the magnitude of $$$ {\theta}_{c, noOC} $$$ is only slightly smaller in most industries, we interpret these differences mainly to be due to the first explanation.

Robustness Check (4): Investment in Intangible Assets

The fourth robustness check tests the sensitivity of our results to the use of intangible capital stock. Our main specification follows Corrado et al. (2005) and Corrado et al. (2009), by capitalizing intangible expenditure to account for the fact that the effect of intangible capital is cumulative and affects firm performance over multiple years. Corrado et al. (2005) and Corrado et al. (2009) argue that intangible capital should be treated analogously to tangible capital. However, this can be questioned if intangibles are not used as production inputs but are modeled to govern productivity. Moreover, the literature regularly uses expenditure or investment flows for intangible assets instead of capital stocks (see Table A.2). We therefore estimate our model with intangible investment flows as opposed to stocks. The resulting average marginal effects, $$$ {\theta}_{c, Invest} $$$, are positive and significant in most industries, but also of a smaller magnitude (see the seventh and eighth columns of Tables A.6 and A.7).

This is in line with our assumption that intangible capital has a beneficial effect over multiple years and supports the results of the main specification. The industries that seem most sensitive to this measure are the same ones flagged by the other robustness checks.

Heterogeneity in the Elasticity of Intangible Capital

Figure 4 illustrates the heterogeneity in the elasticity of intangible capital for the main specification and each of the four robustness checks. Each line represents the distribution of the quantity $$$ {\overline{\theta}}_c^{Q_T}-{\overline{\theta}}_c^{Q_B} $$$ (in the main specification this is reported in the third and seventh columns of Table 2). The thick solid line shows the distribution of differences from the main specification. The thin solid line corresponds to the estimates based on a translog production functions (robust spec. [2]), the dotted line corresponds to the estimates without organizational capital (robust spec. [3]), and the dashed line corresponds to the estimates using investment flows (robust spec. [4]).28

It emerges from Figure 4 that using a translog production function instead of a Cobb–Douglas specification, while maintaining a nonlinear specification for the law of motion in both settings, does not have much of an effect on the heterogeneity of $$$ {\theta}_c $$$, as most differences remain positive. The third and fourth robustness checks generally yield coefficients of smaller magnitude, which explains why the distributions are shifted to the left. These distributions also show less weight in the right tail, suggesting lower kurtosis. However, in both these cases, the majority of the differences are still positive.29 In other words, the average $$$ {\theta}_c $$$ in the top quintile is higher than the average $$$ {\theta}_c $$$ in the bottom quintile in the majority of industries. This is consistent with the pattern shown in Figure 3 and confirms not only the heterogeneity of the impact of intangible capital on TFP, but also the fact that firms with a larger stock of intangible capital also benefit more from continued investment in this type of asset.

6 Productivity Dynamics at the Industry Level

Next, we present evidence relating to the second research question of this analysis, linking the use of intangible capital at the firm level with industry trends in productivity divergence. We adopt the approach of Andrews et al. (2019) and compare the productivity performance of two groups of firms: a group of top performers, called “frontier” firms, and a representative group of all other firms, called “followers.” Following these authors, we define the intangible frontier as the group of firms with the top 5 percent largest stocks of intangible capital in each industry and year. For ease of comparison with the results of Andrews et al. (2019), we also define a TFP frontier as the top 5 percent most productive firms in each industry and year. With this methodology, the identity of firms that populate the frontier group differs each year. Thus, our results should be interpreted as representing changes in the upper tail of the distributions, of intangible capital or TFP, rather than the behavior of a representative firm in the respective group.

6.2 Aggregate Productivity Dynamics

Figure 5 depicts the evolution of productivity relative to the base year (2009) in the manufacturing sector. For each of three groups of firms, those on the intangible frontier (dashed line), those on the TFP frontier (dotted line), and followers (solid line), we plot the difference between productivity in year $$$ t $$$, and productivity in the base year.31 Figure 6 provides the equivalent picture for the services sector, starting in the base year 2008.32 Both these figures suggest that firms on the intangible frontier have outperformed the industry average over a 5-year period, whereas this is not the case for firms on the TFP frontier.

In the manufacturing sector, the productivity of the average firm grew by 6 percent over the period 2009–2014. The productivity of firms on the intangible frontier grew by 9 percent, resulting in a cumulative productivity gap of 3 percentage points over these 5 years. On the contrary, the productivity of firms on the TFP frontier grew by 5.3 percent, closely following the evolution of the industry average. In the services sector, the productivity of the average firm fell sharply between 2008 and 2009, and did not recover by 2013, when productivity was still 6.5 percent lower than in 2008. The productivity of firms on the TFP frontier fell even more sharply, remaining 8 percent lower in 2013 than in 2008, which results in a negative cumulative gap of 1.5 percentage points compared to the average firm. On the contrary, firms on the intangible frontier managed to recover some of the decline, and their productivity was only 2 percent lower in 2013 compared to 2008. Firms on the intangible frontier therefore grew 4.5 percentage points faster than the average firm over the 2008–2013 period.

The lack of divergence between firms on the national TFP frontier and follower firms contrasts with the result of Andrews et al. (2019).33 However, our results for Germany are consistent with those of Schiersch (2019), who tests different measures productivity and those of Lehmann and Keilbach (2019). It should be noted that even though Figures 5 and 6 do not show evidence of divergence in TFP growth along the TFP frontier, there is still divergence in TFP levels. In the respective base years, the TFP level on the frontier was twice the magnitude of those of followers in the manufacturing sector (base year 2009). In the services sector, it was four times larger on the frontier (base year 2004) than for followers. Given the similar growth rates of TFP in both frontier and follower groups, divergence in the level of TFP has increased, a result in line with Andrews et al. (2019).

The presence of divergence along the intangible frontier supports the hypothesis that intangible capital is a likely driver of this trend. To assess the sensitivity of this result to the estimation strategy, Figures 7 and 8 plot the evolution of the difference in log productivity levels between firms on the intangible frontier and followers for each of the four robustness specifications discussed in Section 5.2 (described in Table 3). In other words, they show the development of the productivity gap between the frontier and the rest. The direction and size of the productivity gap hold across the different specifications.

Furthermore, we plot the evolution of the productivity gap between followers and frontier firms using two additional definitions of the frontier: TFP and intangible intensity. The former simply visualizes the results discussed earlier, emphasizing that the cumulative difference is negative over a 5-year period. The latter is defined as the ratio of intangible capital stock over physical capital stock. This tests the sensitivity of the result to the definition of the intangible frontier. In particular, with a frontier defined in terms of stocks, firms with small stocks that might nevertheless be very knowledge intensive would not be classified as belonging to the frontier. Figure 7 confirms that in manufacturing the evolution of productivity is similar for firms with high intangible intensity as that of firms with large intangible stocks. Figure 8 suggests that, in services, the cumulative gap in productivity between frontier firms and followers is even more marked when the frontier is defined in terms of intangible intensity. This strengthens our interpretation that intangible capital plays a role in driving productivity divergence.

7 Conclusion

This study contributes to the discussion around the global slowdown of aggregate productivity growth and the accompanying increase in productivity divergence between top performers and the rest. Although these developments are well documented, the underlying drivers are still not well understood and the subject of ongoing debate, among both academics and policy-makers. The present study examines the role of intangible capital in this process. The literature clearly establishes that intangible investment has a positive impact on the productivity of the average firm, which makes the deceleration of productivity growth all the more puzzling. However, models in the endogeneous growth literature highlight the mechanisms through which heterogeneity in the efficiency with which firms deploy intangible capital can exacerbate productivity differences between firms, with negative consequences for long-term innovation and growth.

Thus, the present paper explores two empirical research questions. First, we test whether the elasticity of intangible capital on productivity is indeed heterogeneous across firms and whether it increases with the size of a firm’s stock of intangible capital. Second, we test whether these firm-level results are linked to industry-level trends in productivity divergence and growth.

The estimation results confirm the heterogeneity of the relationship between intangible capital and productivity. Within industries, the marginal effect of intangible capital increases in magnitude with the size of the stock of intangible capital and is highest for the group of firms located at the top of the intangible capital distribution. In the bottom quintile of the intangible capital distribution, the marginal effect is often not different from zero or even negative in a number of industries. In other words, firms that manage to build up a sizeable stock of intangibles are able to outperform their peers, whereas firms with small or no stock of intangibles have little incentive to start investing, as these investments will not yield interesting returns. Therefore, firms do not seem to benefit uniformly from investments in intangible capital, rationalizing the observed skewness of the distribution of intangible investment across firms.

Previous studies document, using international data, a growing gap in productivity growth between firms in the top of the productivity distribution and the rest. We extend this discussion by exploring the role of intangible capital in productivity divergence and relate our firm-level estimation results to the evolution of productivity at the aggregate level. We find that firms with the top 5 percent largest stocks of intangible capital have seen faster productivity growth than the population of firms. The gap in productivity growth amounted to 3 percentage points in manufacturing and 4.5 percentage points in services over a 5-year period. We also find increasing entrenchment on the frontier, symptomatic of declining business dynamism.

Additional analyses at the detailed industry level reveal a positive correlation between, on one hand, the gap in productivity growth between frontier firms and followers, and on the other hand, differences between these groups in the marginal effect of intangible capital. These findings suggest that the heterogeneous effect of intangible capital observed at the firm level translates into growing productivity divergence at the sector level. Furthermore, we show that industries in which the productivity gap between firms with large stocks of intangibles and the rest was more marked are also those industries in which average productivity growth was lower, supporting the role of intangible capital as a driver of slow productivity growth.

The results of the present analysis pinpoint intangible capital as a factor behind productivity divergence and low growth, raising the question of whether policy interventions could narrow the investment gap between frontier firms and the rest. The heterogeneity in returns to intangible investment rationally explains the observed investment gap, but the design of effective policies requires a clearer understanding of the causes behind this heterogeneity. Of particular interest are the poor returns obtained by firms who invest only small amounts, and the existence of a minimum threshold of investment. These differences are typically modeled as exogenous (see Aghion et al., 2019; De Ridder, 2023), but a number of potential explanations emerge from the literature.

Bloom et al. (2013) and Brynjolfsson et al. (2021) argue that the effective roll out of a new general purpose technology (GPT), such as ICT or artificial intelligence, requires complementary investments in organizational practices and human capital. If some firms invest only in a subset of the required assets, they will be unable to reap the full benefits of their investments. Future research should further clarify which assets are particular bottlenecks in firms’ investment strategies. Moreover, complementarities exist between firm investments and physical digital infrastructure. Barriers to access can explain the low returns to firm investments, thus justifying policy intervention in the build-up of, and ensuring fair access to, this infrastructure.

The presence of spillovers arising from knowledge assets is a likely driver of both the low returns at the bottom of the distribution and the high returns at the top of the distribution. Managing knowledge assets requires specific internal capabilities and skills to ensure both an effective protection of intellectual property rights and the absorption of knowledge from external sources. As there are a number of potentially important policy implications, future research should explore the relationship between the distribution of these skills across firms and the heterogeneity in the returns to intangible investment. On the one hand, if large firms are in a privileged position to absorb knowledge from other firms, differences in intangible investment will be further exacerbated. On the other hand, if small firms are vulnerable to having their investments appropriated by competitors, they will have fewer incentives to invest, even if they can obtain a positive boost to their productivity. Thus, strengthening the intellectual property regime to the benefit of smaller intangible investors can be a tool to remedy the investment gap.

The entrenched gap in intangible investment between frontier firms and the rest can be driven by other factors that limit the ability of laggards to invest, beyond the low returns documented in Section 5. Notably, evidence suggests that investing in intangibles requires different financial instruments compared to investing in physical capital and that some firms might face financial constraints (Haskel and Westlake, 2018). Further research is needed to quantify the importance of financing constraints, to understand which characteristics of intangible assets create challenges for the financial system and to design policy remedies.

Finally, much remains to be explored to fully understand the interaction between the concentration of intangible investment in a limited number of firms and its impact on the contestability of markets and on the development of market power. In particular, the relationship between intangible investments and the ability of firms to charge mark-ups is complex. On the one hand, firms need to charge prices above marginal costs to recoup their spending on these assets. Furthermore, intangible investments enable firms to produce higher quality goods and services, thereby differentiating themselves from competitors. On the other hand, intangible investments allow firms to escape competition and gain market power, which might come at the detriment of follower firms and consumers. The role of policy interventions, such as product market regulations or the enforcement of competition policy, is central to ensuring fair market access to successful innovators and to avoid reinforcing the entrenchment dynamics created by intangible capital.