the rise of intangible capital and the macroeconomic
TRANSCRIPT
The Rise of Intangible Capital and the MacroeconomicImplications*
Andrea Chiavari† Sampreet Goraya‡
November 13, 2020
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Abstract
Intangible capital, which includes investments in research and development, and intellec-tual property product, has risen dramatically in the last decades, however little is knownabout its importance in production. We estimate a standard production function aug-mented with intangible capital and show that its input share has tripled since the 1980sand this increase has come at the expense of labor in production. We label this finding asIntangible Capital Biased Technological Change. Further, we provide empirical evidence con-sistent with intangible capital being significantly more sunk, meaning that it entails higherinvestment adjustment costs than traditional capital. Finally, due to the presence of highadjustment costs, we show that intangible capital tends to be misallocated (i.e., the averageproduct of intangible capital is more dispersed than the average product of traditional cap-ital). Using a structural framework of firm dynamics, we show that a shift in productionfrom a flexible input – labor – to an input with high adjustment costs – intangible capital –can explain a sizable fraction of some of the major trends documented for the US economy,such as the rising average firm size, dispersion of total factor productivity revenue, sales-weighted profit rates and increasing industry concentration.
Keywords: Intangible Capital, Adjustment Costs, Production Function, Lumpiness, Misal-location, Concentration.
JEL Codes: D24, D25, E22, O34
*We are grateful to Isaac Baley, Julian di Giovanni, Manuel Garcia Santana, Edouard Schaal and AlessandroTarozzi for their invaluable support. We sincerely thank Jan Eeckhout for long discussions that substantially im-proved the paper. We also thank Vladimir Asriyan, Rudi Bachmann, Fernando Broner, Andrea Caggese, GiacomoCarlini, Davide Debortoli, Jan De Loecker, Maarten De Ridder, Jordi Galí, Andrea Fabiani, Javier Gomez Biscarri,Priit Jeenas, Raffaele Manini, David Nagy, Giacomo Ponzetto, Maria Sandström, Venky Venkateswaran, JaumeVentura as well as the participants at CREI international lunch, CREI macro lunch, NuCamp Virtual Workshop2020 and European Economic Association Congress 2020 for their helpful comments and discussions.
†Department of Economics, Universitat Pompeu Fabra and Barcelona GSE. Email: [email protected]‡Department of Economics, Universitat Pompeu Fabra and Barcelona GSE. Email: [email protected]
1 Introduction
In the last decades, firms’ investments in research and development, intellectual propertyproduct, and computerized information – commonly known as intangible capital – have risensignificantly as measured by their share in total capital.1 However, because of its intangiblenature, it is inherently difficult to evaluate how intangible capital interacts with other inputsof production, and how its emergence has changed the nature of production technologies. Theaim of this paper is to answer the following questions: Is intangible capital an input in pro-duction and what is its input share? What are the main characteristics of intangible capitaland how do they shape firms’ investment behavior? What are the implications of the rise ofintangible investments for the US economy? In doing so, we provide a unified explanationfor some of the major trends that have been witnessed in the US economy and have caughtthe attention of both academic researchers and policy makers, such as the increasing firm size,misallocation of resources, industry concentration, and sales-weighted profit rates.2
We begin by documenting the changing nature of the production technology. We showthat intangible capital is a dynamic input in production and provide a precise estimate of itsinput share by using production function estimation techniques developed in the empiricalindustrial organization literature. Further, we show that the input share of intangible capitalhas witnessed a threefold increase since the 1980s, and this has come at the expense of the laborinput in production. We label this phenomenon Intangible capital Biased Technological Change(IBTC from here onwards).
Thereafter, we provide novel empirical evidence on the nature of the intangible invest-ment process. We document that the intangible investment rate distribution is characterizedby higher inaction rates (i.e., periods of zero investment), and greater dispersion and serialcorrelation relative to physical capital. In order to rationalize these empirical findings, we em-bed intangible capital into a quantitative framework of firm and investment dynamics withadjustment costs associated with both capitals. The model estimates much higher adjustmentcosts – particularly fixed adjustment costs – for intangible capital investments relative to thatof physical capital investments. These findings are in line with the nature of intangible capital,e.g., consider the case of the Just In Time (JIT) production process.3 This is commonly used bymanufacturing firms nowadays, and requires large investments beforehand due to the pres-ence of inherent indivisibilities in the investment process. Further, implementation of thesetechniques takes long set up times, such as workers’ retraining and restructuring of businessprocedures, which makes investment in intangible capital a costly endeavor for firms.
Finally, we use the estimated model to quantify the effects of IBTC on firm size, industryconcentration, profit rates and on the allocative efficiency of the economy. We show that, a
1For instance, see Corrado, Hulten, and Sichel (2009), McGrattan and Prescott (2010) and Koh, Santaeulalia-Llopis, and Zheng (2016).
2For increasing profit rates and concentration see Barkai (2016), Autor, Dorn, Katz, Patterson, and Van Reenen(2017), De Loecker, Eeckhout, and Unger (2020), and Grullon, Larkin, and Michaely (2019). For increasing misal-location see Kehrig (2015), Caggese and Perez-Orive (2017), Hsieh and Klenow (2017) and Decker, Haltiwanger,Jarmin, and Miranda (2018).
3Just In Time (JIT) production techniques, pioneered by Japanese manufacturers, gained momentum amongU.S. manufacturers in the early 1980s. Nakamura, Sakakibara, and Schroeder (1998) wrote “Because so manydifferent aspects of plant operation are involved, the transfer of JIT requires a substantial effort on the part of U.S.manufacturers” and Fullerton, McWatters, and Fawson (2003) wrote “Investment returns from JIT adoption are notimmediately observable, due to the long-run nature of its implementation process.”
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shift in production technology towards intangible capital – that is costly to adjust – increasesthe threshold of productivity of firms above which they get involved in production, which inturn increases the average productivity of incumbent firms. Further, this implies an increasein average size and concentration of economic activity. Moreover, intangible capital tends tobe more misallocated as it does not reallocate quickly enough from low productivity to highproductivity firms. Therefore, as intangible capital becomes more important as an input ofproduction relative to physical capital and labor, the overall misallocation (as measured by thedispersion in Total Factor Productivity Revenue, TFPR) in the economy increases as well.
Our results imply that a significant part of the increase in misallocation of resources, firmsize, concentration and the sales-weighted profit rates is a result of the changing nature ofthe production technology and does not imply any deviations from the first-best allocationin the economy. This is in contrast to several recent contributions that have focused on thedistortionary consequences of the technological change. Therefore, our results caution thepolicymakers to have a broader view of the changes that we witness in the US economy.
For our stylized facts, we exploit data on US publicly traded firms from Compustat, wherewe can observe different expenditure invoices reported by the firms and their balance sheets aswell. Measuring intangible capital is a difficult task at any level of aggregation as accountingstandard (US GAAP in the case of U.S.) are insufficient to satisfactory book the intangiblecapital on the balance sheets.4 To do this end, we follow the methodology in Peters and Taylor(2017) and Ewens, Peters, and Wang (2019) to construct our baseline measure of intangiblecapital. In particular, it consists of two main components: First, we use the Perpetual InventoryMethod (PIM from here onwards) to convert internal intangible investments (e.g., expenditurein research and development) into the stock of internally generated intangible capital.5 Second,we use the stock of externally acquired intangible capital that is directly available in the firms’balance sheet. A sum of these two stocks gives us our firm-level measure of intangible capitalbetween the years 1980 and 2010. However, we recognize that there is no perfect measureof intangible capital. Therefore, to test the robustness of our baseline measure, we comparewith alternative firm-level measures and with aggregate-level measures provided by BEA andCorrado, Hulten, and Sichel (2009).6
We make three contributions in this paper. First, using the data mentioned above, weestimate a firm-level production function with three inputs, namely, intangible capital, phys-ical capital, and labor. This is well known to be a difficult task because firms can observetheir productivity while choosing their inputs, but the researcher cannot. We tackle this en-dogeneity problem by leveraging various estimation techniques. In particular, we estimatethe Cobb-Douglas production function using cost shares, a dynamic panel data approach, theOlley-Pakes estimator (Olley and Pakes, 1992) and finally, the Ackerberg-Caves-Frazer estima-tor (Ackerberg, Caves, and Frazer, 2015). We use a rolling-window estimation to allow fortime-varying input shares.7
4See, Lev and Gu (2016) and Ewens, Peters, and Wang (2019) that highlights the problems with US GAAP, andCorrado, Hulten, and Sichel (2009) that documents the inability of National and Income Accounts (NIPA tables) tofully capture the importance of intangible capital in the economy.
5In such cases, the direct capitalization of intangible investment is not allowed by the law. This is due to thefact that, for internally produced intangible capital, there is no reference market price, which makes it difficult toproperly value it in the firms’ balance sheet.
6For detailed discussion on these issues, we refer to the Online Appendix.7We assume Cobb-Douglas functional form as our baseline case. However, this choice of functional form is
restrictive because it does not allow any non-linearity of or any interactions among inputs. Hence, we also consider
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All of these methods find that intangible capital was already an important input of pro-duction at the beginning of the sample period, as its output elasticity ranges from 0.03 to 0.09in the sample period 1981-1990. Then, we find that the intangible capital input share changedover time, tripling between the year 1980 and 2007 in our preferred estimation technique (i.e.,the one performed using the Olley-Pakes estimator with our baseline measure of intangiblecapital).8 Moreover, this was accompanied by a substantial decline in the labor input share,suggesting a degree of substitution between these two inputs.
Moreover, our results suggests that omitting intangible capital in the production functionestimation will generate an omitted variable bias. Further, this bias will be increasing over timeas the input share of intangible capital rises and input share of labor declines. This implies thatan increasing part of the variation in sales will be attributed to firms’ productivity and thus,it will deteriorate our inference on estimated input shares, returns to scale and productivitydistribution.
Second, we build a structural framework of firm and investment dynamics, where firmsproduce using a Cobb-Douglas production function with intangible capital, physical capitaland labor. Moreover, the model features both endogenous entry and exit of firms and a richand flexible structure of investment adjustment costs for both capitals. In particular, invest-ment adjustment costs have two parts: a convex part that disciplines the intensive margin offirm investments, and a fixed part that disciplines the extensive margin of firm investments.The model allows us to precisely pin down the differences in the investment process of thetwo capitals, and it enables us to evaluate the macroeconomic implications of the IBTC (i.e.,the changing nature of the production technology) in the US economy.9
The quantitative predictions of the model depend on the identification of two sets of param-eters that describe the convex and the fixed cost of adjustment for both physical and intangiblecapital. Following the seminal paper of Cooper and Haltiwanger (2006), we use inaction rates,defined as investment between ±1%, to identify fixed costs of adjusting each type of capital.This is done because higher fixed costs of adjusting capital increase the inaction rate in themodel, as firms prefer not to invest instead of paying these costs. Moreover, we use the auto-correlation of the investment rate to discipline the convex costs, since high convex adjustmentcosts force the firms in the model to build up their capital more slowly, making investmentsmore serially correlated over time. It is important to note that all of these parameters are jointlyestimated in the steady state.10
The estimated model broadly captures the cross-sectional heterogeneity in firm size, ageand intangible intensity (measured as stock of intangible capital over employment), and fur-ther matches well the lifecycle of firms as observed in the data. It estimates substantial differ-ences in the investment process of these two types of capital, with intangible capital entailing
the case of the Translog form to address this issue.8We select the Olley-Pakes estimator as our preferred measure since it has been the most widely used in recent
papers based on Compustat, e.g, Bloom and Van Reenen (2007), Imrohoroglu and Tüzel (2014) and De Loecker,Eeckhout, and Unger (2020). We also estimate production function with several other measures of intangible capitaland found that all these estimations give us an increasing trend in the intangible input share over time (see, theOnline Appendix).
9The estimation of the production function in the first part of the paper is consistent with the structure of themacro-model. In particular, the production function estimation takes into account the quasi-fixed nature of certaininputs, such as physical and intangible capital.
10High fixed costs decrease the autocorrelation of the investment rates, whereas, the convex costs increase it -these two counterbalancing forces are crucial for the correct identification of these two costs.
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much higher adjustment costs compared to physical capital – particularly, fixed adjustmentcosts. This captures, to a certain extent, the notion of sunkness as described in Haskel andWestlake (2018):
If a business makes an intangible investment and later on decides it wants to back out, it’soften hard to reverse the decision and try to get back the investment’s cost by selling thecreated asset – and in general, it’s harder than in the case of a tangible asset. [. . . ] The [. . . ]reason tangible investments are easier to sell is that they are less likely to be uniquely linkedto the firm that owns them and its business.
The fact that high adjustment costs are linked to intangible capital implies that the firmswill be slower at adjusting their intangible capital relative to physical capital when they arehit by a productivity shock. To validate this model prediction, we estimate the elasticity ofthe average revenue product of both forms of capital to idiosyncratic productivity shocks. Wefind that this elasticity is much higher in case of intangible capital, making the average revenueproduct of intangible capital, ARPKi, much more volatile relative to the average revenue prod-uct of physical capital, ARPKp, at the firm-level. This finding suggests that high adjustmentcosts make the reallocation of intangible capital slower relative to the one of physical capitalacross firms, implying that intangible capital is more misallocated relative to the other inputsof production in the cross-section. We show that the model is able to replicate the distribu-tion of ARPKi and ARPKp as documented in the data and that the ARPKi is more dispersedrelative to the ARPKp in all sectors.
In our third contribution, we assess the quantitative importance of the IBTC in explainingsome of the recent trends in the US economy. To do that, we rely on our baseline measure ofintangible capital and use the input shares provided by Olley-Pakes estimation.11 We find thatthe rising importance of intangible capital in production between 1980 and 2007 can explainaround one-fifth of the increase in the average firm size, one-third of the rise in the disper-sion of TFPR, and around two-thirds of the rise in sales-weighted profit rates and industryconcentration in the same time period.
These results follow directly from the nature of the technological change. As produc-tion technology becomes more intangible-intensive, firms invest more in an input that entailshigher adjustment costs, as a result, the value of entry decreases and average productivity ofentrants increases. This implies that a relatively small – but more productive – mass of firmsoperate in the economy; thus, the average incumbent size increases. Furthermore, due to thenature of adjustment costs, we document a reallocation of economic activity from small firmstowards large firms. This happens because, a rise in adjustments costs (fueled by the rise inthe intangible input share), makes it more difficult for small firms to grow – particularly newentrants. However, large (high productivity) firms, as they expect lower productivity in the fu-ture, due to a mean reverting productivity process, have an outside option of letting the capitalstock to depreciate without incurring any additional cost of adjusting. A decline in the mass offirms and size-biased effect of the adjustment costs implies higher concentration in the econ-omy and a decline in labor demand, that in turn results in lower equilibrium wages. Moreover,profit rates increase due to a combination of multiple forces: on the one hand, greater depen-
11It is important to note that the estimated quantities on macroeconomic implications may vary depending uponthe measure of intangible capital used and the production function estimation technique. However, the resultswould remain same qualitatively.
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dence on intangible capital decreases firms’ profits. On the other hand, better selection of firmsin combination with lower wages increase profits for the average firm, and the latter force pre-vails in the quantitative exercise.
Furthermore, in an intangible-intensive economy, firms use a production technology thatis more dependent on an input that is highly distorted (i.e., the distribution of ARPKi is highlydispersed) due to the presence of high adjustment costs; therefore, the overall allocative effi-ciency of the economy decreases (i.e., the dispersion in TFPR increases). This happens becauseTFPR in our framework is a weighted geometric mean of the average revenue product of in-puts, where the weights are proportional to their output elasticities. Due to the presence ofadjustment costs, dispersion in TFPR is driven by dispersion in average products of both cap-itals. Therefore, when an economy shifts from labor to intangible capital, the dispersion inARPKi becomes the primary driver of the dispersion in TFPR. A well established literature,pioneered by Hsieh and Klenow (2009), has interpreted the dispersion in TFPR as a sign ofmisallocation of resources in the economy, as predicted by a static frictionless model. How-ever, in our dynamic framework, this dispersion is generated by technological constraints inthe investment process, as explained in Asker, Collard-Wexler, and De Loecker (2014), andcannot be interpreted as a loss of efficiency. The equilibrium allocation in our model is efficientand corresponds to the optimal allocation provided by the social planner.
Having assessed the macroeconomic implications of the IBTC, we document the ability ofthe model to capture cross-sectional heterogeneity in the intangible intensity and its sectoralimplications. We find support for the main predictions of the model. In particular, the share ofintangible investment over output increases in the intangible intensity. Further, in intangible-intensive sectors, TFPR distribution is characterized by high mean and dispersion. Finally,these sectors exhibit high concentration of productive activity and larger profit rates.
Literature Review. This paper is related to the literature that measures intangible capital atthe aggregate level as in Corrado, Hulten, and Sichel (2009), McGrattan and Prescott (2010) andKoh, Santaeulalia-Llopis, and Zheng (2016), and at firm-level as in Peters and Taylor (2017) andEwens, Peters, and Wang (2019). Similar to us, Eisfeldt and Papanikolaou (2013), Ayyagari,Demirguc-Kunt, and Maksimovic (2018) and Belo, Gala, Salomao, and Vitorino (2019) showthat intangible capital is an important source of firm’s value and performance. Relative tothem, we structurally estimate a production function with three inputs: intangible capital,physical capital and labor.12 We document that intangible capital is an important input ofproduction which is rising over time at the expense of labor. In particular, these results at thefirm-level echoes the findings of Corrado, Hulten, and Sichel (2009) and Koh, Santaeulalia-Llopis, and Zheng (2016) at the aggregate-level.
Furthermore, our paper is related to the extensive literature that examines lumpy invest-ment dynamics pioneered by Abel and Eberly (1994), Abel and Eberly (1996), Doms and Dunne(1998) and Cooper and Haltiwanger (2006), which highlights the role of non-convex adjust-ment cost in the firm-level investment process. To the best of our knowledge, we are the firstones to highlight the fact that the presence of adjustment costs are an important feature of theintangible investment process and estimate them using a structural framework. Further, we
12Moreover, our results suggest a degree of complementarity between both capitals. Using a different frame-work, Eisfeldt, Falato, and Xiaolan (2019) also show that intangible investments (embodied as human capital)complement physical capital.
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highlight stark differences in the adjustment costs of both capitals. These results are consis-tent with Sun and Xiaolan (2019), and Döttling, Perotti, and Ladika (2019), that argue that thecreation of intangible capital requires skilled human capital that is costly to adjust.13
Our paper is also related to Kehrig (2015), Hsieh and Klenow (2017) and Decker, Halti-wanger, Jarmin, and Miranda (2018), that document an increase in the dispersion of TFPR.Moreover, it is related to Barkai (2016), Diez, Fan, and Villegas-Sánchez (2019) and De Loecker,Eeckhout, and Unger (2020) that document a rise in profit rates, and to Autor, Dorn, Katz,Patterson, and Van Reenen (2017) and Grullon, Larkin, and Michaely (2019) that show the risein industry concentration in the US. Gutiérrez and Philippon (2017), and Crouzet and Eberly(2019) document a potential link between the rise of intangible capital and both the increasingindustry concentration and the weakening investment in physical capital.
This paper is related to Karabarbounis and Neiman (2019) that argue that unmeasured cap-ital also explains the rise of factorless income (i.e., excess profits in our case). Consistent withthese results, we find that the rise in accounting profits is smaller in our baseline specifica-tion, where we include intangible capital as an input in production, relative to the one that ismentioned in De Loecker, Eeckhout, and Unger (2020) and Barkai (2016).
Our results complement the findings of Caggese and Perez-Orive (2017), as in their case,indivisibilities in investment process of intangible capital are required for collateral constraintsto be binding at the margin. Further, Döttling and Ratnovski (2020) mention that intangibleinvestment may respond less to monetary policy shocks because of higher investment adjust-ment costs and our paper provides direct evidence on these costs.
Finally, recent papers have presented potential mechanisms behind different macroeco-nomic trends and the rise of intangible capital, where intangible investments are assumed tobe a fixed cost for firms. Korinek and Ng (2018) argue that digital innovations, that requirea fixed investment but can be reproduced at zero marginal cost can explain the rising marketconcentration. Weiss (2019) presents a model with oligopolistic competition and shows howintangible capital can explain the rise of market power. De Ridder (2019), that uses a frame-work of endogenous growth, shows how a rise in fixed costs can explain the decline in pro-ductivity growth and the rise of markups, whereas, Hsieh and Rossi-Hansberg (2019) theorizehow increasing fixed costs and scalability can be behind the divergence between local marketconcentration and aggregate market concentration. Aghion, Bergeaud, Boppart, Klenow, andLi (2019), using an endogenous growth framework, argue that IT investment lowers the costsof running multiple plants that increases market concentration. Zhang (2019) shows that a de-cline in the price of intangible capital can explain both the rise in concentration and the declinein the labor share. Lashkari, Bauer, and Boussard (2018) argue that IT investments raise thereturns to scale at the firm level and this could explain the rise in concentration among Frenchfirms. Sandström (2020) argues that intangible investments increase uncertainty of returns thatmay increase markups but not profits. Relative to all of these papers, we inform our way ofmodeling the intangible investment process by the empirical evidence that is based on firm-level measure of intangible capital. The empirical evidence suggests that intangible capital isa dynamic input, while its investment process is very lumpy.14 Combining these insights with
13In these papers, intangible capital is the result of a joint investment effort by firms and highly skilled employ-ees. The costs to hire and fire skilled employees, that includes search costs, are absorbed by our estimates of theadjustment costs.
14Interpreting intangible investments as fixed costs that firms have to pay every period would impose strict
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a structural model, to the best of our knowledge, we are the first ones to provide a unified ex-planation for the rise in misallocation in conjunction with other macroeconomic trends, suchas the rise of industry concentration, profits and firm size.
A related literature explains the macroeconomic trends with a different mechanism. Aki-cigit and Ates (2019b), and Akicigit and Ates (2019a) highlight the role of declining knowledgediffusion in explaining the rising concentration in the US economy. Finally, Hopenhayn, Neira,and Singhania (2018) and Peters and Walsh (2019) argues that the decline in the labor forcegrowth rate can explain the increase in concentration and decline in the business dynamism.
Outline. Section 2 briefly discusses data and shows the construction of our main variables.Section 3 documents the stylized facts. Section 4 presents our quantitative framework. Section5 contains the structural estimation of the model and its external validation, whereas, section 6presents the main results and discusses the implications of IBTC for the US economy. Section 7discusses the model assumptions and its implication for the main results and provides furtherrobustness checks. Section 8 concludes.
2 Data and Measurement
In this section, we present the construction of our empirical measures which rely upon the pub-licly available US Compustat database from the years 1965-2010. We describe the constructionof intangible capital and physical capital using data from financial statement and balance sheet.We use these variables to estimate the production function for the whole economy.
2.1 Main Measures
Here, we present the construction of our main empirical measures, particularly how we con-struct our time-series of intangible capital. The US Compustat is a firm-level database with allthe US publicly traded firms between 1965 to 2015. However, we focus our analysis from 1980onward, since this is the period around which intangible capital has started to grow and theaccounting standards have fully stabilized. The choice of this data is motivated mainly by itsability to span the longest possible time period and the largest number of sectors. Despite thefact that publicly traded firms are few relative to the total number of firms, and they tend tobe the largest firms in the economy, they still account for almost 30% of US employment, asreported by Davis, Haltiwanger, Jarmin, Miranda, Foote, and Nagypal (2006).
The Compustat data contains information on firm-level financial statements. Particularly,we observe measures of sales, input expenditures, capital stock information, as well as detailedindustry activity classification. In appendix A, we explain in detail the cleaning process of theraw data and how we deflate the variables.
We use Perpetual Inventory Method (PIM) to estimate the stock of physical capital at firm-level. Particularly, we use the following equation:
kp, f t = (1− δp)kp, f t−1 + xp, f t,
assumptions on the nature of the intangible investment process. First, the depreciation rate of intangible capital ishundred percent and thus, it is a static input. Second, intangible investments should be strongly correlated overtime. However, data inform us that intangible investments are lumpy and the depreciation rate of intangible capitalis higher than that of physical capital but it is far below hundred percent.
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where xp, f t − δpkp, f t−1 = PPENTf t − PPENTf t−1 is net investment in physical capital kp byfirm f in year t, where PPENT is defined as the net value of property, plant, and equipment.The initial stock of physical capital is initialized using the first available entry of PPEGTf t,defined as the gross value of plant, property and equipment. Since we want a measure of realcapital stock, we deflate our measure of net investment with the investment price deflator asprovided by the BEA.
2.2 Intangible Capital: Accounting Standards and Measurements
Measuring intangible capital is a difficult task as a substantial portion of it is internally gener-ated, instead of being externally acquired, unlike physical capital because modern accountingstandards (US GAAP) do not allow the capitalization of internally generated intangible capi-tal on firms’ balance sheets (see, Lev and Gu, 2016 and Ewens, Peters, and Wang, 2019). Fornearly all the internally generated intangible capital, such as knowledge and organizationalcapital, accounting standards differ significantly from physical capital.15 All physical invest-ment is recorded on the balance sheet at its purchase price and depreciated over its useful life.However, internally produced intangible investment, such as R&D, advertising, or employeestraining is fully expensed in the current period and therefore, appears in the firms’ incomestatement. Instead, externally acquired intangible capital is directly booked on firms’ balancesheet.
In light of these considerations, our main measure in the paper is formed by two differentcomponents: The first one is the internally generated intangible capital, which in our case is ob-tained through the capitalization of R&D expenditure via a perpetual inventory method withsector-specific depreciation rates and an appropriate Intellectual Property Product (IPP) defla-tor. We do not include organizational capital in our benchmark measure for three reasons.16
First, organizational capital is constructed through the capitalization of a sector-dependentfraction of Selling, General, and Administrative expenses (SG&A). This item includes manyexpenditures that are not inherently related to intangible capital like CEO wages, rents forbuildings, and capital adjustment costs, among others.17 This raises the concern that capital-izing adjustment costs as measured capital would artificially inflate our measure of intangiblecapital and would also create conceptual issues for the estimation of the production function.Second, capitalizing such a big expenditure item would heavily downward bias our estimatesof inaction rate as this expenditure item is never zero and even in periods of no investment inintangible capital we would be capitalizing, for instance, CEO wages and rents for buildings.Third, the imputation of a constant fraction across firms of SG&A as intangible investmentswould substantially increase the concerns related to potential measurement error.
The second components is the externally acquired intangible capital, which is capitalizedon firms’ balance sheet at fair value under the variable INTANi,t in Compustat, according to
15However, there are some exceptions. For example, U.S. GAAP treats the development of computer softwaredifferently from other R&D costs. Following ASC 985 (formerly FAS 2), once a software developer has reachedtechnological feasibility," the developer must capitalize and amortize all development costs until the product isavailable for general release to consumers. https://asc.fasb.org/link&sourceid=SL2313776-111772&objid=
650358716The organizational capital is used in Eisfeldt and Papanikolaou (2013), Peters and Taylor (2017), and Ewens,
Peters, and Wang (2019).17While working with Compustat data, it is standard to assume that the capital adjustment costs are expensed in
SG&A.
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the US GAAP under guidelines provided from ASC 350. However, the externally purchasedintangible capital in many cases is acquired through acquisitions of entire firms and this greatlyinfluences the way it is capitalized on the firms’ balance sheet. Normally, the procedure is thefollowing: (i) Tangible capital are identified and capitalized at the fair value, (ii) then identi-fiable intangible capital are capitalized at the fair value in the balance sheet item INTAN f t,and (iii) the residual value is attributed to unidentifiable intangible capital (synergies, orga-nizational culture, etc.) and is capitalized into the Goodwill Gdwl f t. So our final measure ofexternally acquired intangible capital is INTAN f t = INTANO f t + Gdwl f t.
Therefore, we use PIM on R&D expenditure to recover a firm-level measure of knowledgecapital. Particularly, we use the following equation:
krdi, f t = (1− δi)krd
i, f t−1 + xrdi, f t,
where xrdi, f t = XRD f t is gross investment, deflated by IPP price deflator, in the knowledge
capital, the depreciation rate at sector level, δi, is taken from Ewens, Peters, and Wang (2019)and the initial stock, i.e., when firms enter Compustat, is assumed to be zero.18 Therefore, ourbaseline measure of firm-level intangible capital stock is the sum of INTAN f t, appropriatelydeflated, and krd
i, f t, i.e., ki, f t = INTAN f t + krdi, f t. For a more precise discussion on the treatment
of missing values and outliers see Appendix A.However, as the Goodwill is always computed as the residual value, there are reasons to
believe that part of it is not unidentifiable intangible capital.19 If, for instance, Goodwill partlycaptures firm’s investment in acquiring market power in the future, through an acquisition ofanother firm, this would imply that the observed rise in the stock of Goodwill over time is areflection of increasing M&A activity and not a real increase in the stock of intangible capital.This could be particularly problematic for the production function estimation.20.
We address this concern by deflating intangible capital with the Intellectual Property Prod-uct (IPP from here onwards) deflator from NIPA tables.21 Moreover, we construct variousmeasures of intangible capital that do not contain the Goodwill. In particular, additional toour measures are: (i) baseline measure net of the Goodwill, (ii) sum of knowledge capital andorganizational capital.22 We provide the comparison of all these measures in the Online Ap-pendix.
In Figure A.1 in A, we plot the evolution of the real aggregate intangible investments overreal output between 1980-2015. We document steep rise in the real intangible investment share
18For all our analysis, unless differently stated, we exclude all the observation in the first five years in order toavoid strong dependence in our results from our assumption on the initial condition for knowledge capital. Theresults are similar if we use a different level of initial capital, e.g., investment in the first period divided by itsdepreciation rate.
19However, we want to acknowledge that the corporate finance literature− using more detailed data on mergersand acquisitions − has shown that at least 66% of the firms’ Goodwill is indeed true intangible capital (Ewens,Peters, and Wang, 2019).
20We regress the Goodwill against markups and we find no significant association between the two variables.Additionally, it is important to realize that in our production function estimation that is presented in the nextsection, we provide robustness by controlling for firms’ market power using sales shares as proxy variables assuggested by De Loecker, Eeckhout, and Unger (2020)
21However, this is only available at the aggregate-level and not at the firm-level. We want to emphasize that theinability to obtain firm-level investment deflators affects both the measurement of tangible and intangible capitalequally.
22We construct a measure of organizational capital, following Eisfeldt and Papanikolaou (2013) and Peters andTaylor (2017), i.e., korg
i,t = (1− 0.20)korgi,t−1 + γs
SGAi,tIPP deflatort
).
9
over time in the US economy. This is in line with the findings of Corrado, Hulten, and Sichel(2009), McGrattan and Prescott (2010), Koh, Santaeulalia-Llopis, and Zheng (2016), and Petersand Taylor (2017). Finally, we document the ability of our measures of intangible capital tocapture the trends as witnessed in the aggregate data in the national income and product ac-counts (NIPA from here onwards) provided by Bureau of Economic Analysis (BEA from hereonwards) and by Corrado, Hulten, and Sichel (2009). The firm-level measures do a good job incapturing aggregate patterns, however, they are not identical as expected. The firm-level mea-sures show on average a greater increase relative to the aggregate measures and this we thinkcould be due to two reasons: (i) the Compustat database is a selected sample of very largefirms (listed firms) which are more likely to invest more in intangible capital; (ii) the sectoralcomposition of the overall economy is different from that of Compustat’s economy.
In Figure A.2, we use the BEA sectoral shares to aggregate the intangible investment sharein the Compustat economy. It is clear that our baseline intangible capital measure is in linewith the aggregate data and does not over predict the intangible investment at the firm-level.A more detailed discussion on these issues, with more information at disaggregated level anddifferent segments of intangible and physical investments, is provided in the Online Appendix.
Further, we show the evolution of the intensity of externally purchased intangibles capital(i.e., INTAN f t divided by intangible capital) and the evolution of the intensity of knowledgecapital (i.e., krd
f t divided by intangible capital) in A.3 in Appendix A. It is evident that the rise in-tangible capital is driven by the faster accumulation of externally purchased intangibles capitalwithin a sector.
3 Stylized Facts
In this section, we document three stylized facts. First, we structurally estimate the productionfunction with three inputs: intangible capital, physical capital and labor. Further, we use 10year rolling windows to estimate the changes in the input shares over time and documentthe changing nature of the production technology. Second, we compute the investment ratedistributions for physical and intangible capital and highlight their key properties. Third, wereplicate within our dataset, being consistent with our production function estimation, someof the salient facts that have been documented in the literature such as the increasing firm size,standard deviation of TFPR, increasing concentration and increasing profit rates.
3.1 Fact 1: Intangible Capital Input Share has Tripled since 1980s.
In this section, we seek to investigate the importance of intangible capital as a new factor ofproduction; in order to do so, we estimate a production function with three inputs: physicalcapital, intangible capital and labor. Our estimates confirm that intangible capital is an im-portant factor of production. Then, we investigate its evolution over time, finding that theimportance of intangible capital as a factor of production has witnessed a threefold increaseover the last four decades.
3.1.1 Structural Estimation of Production Function
In this section, we structurally estimate the production function with three inputs: intangibles,physical capital and labor. The baseline estimation is performed with the value added as the
10
measure of firms’ output. We follow Bloom and Van Reenen (2007) and Imrohoroglu andTüzel (2014) to compute the value added and wage bill (see Appendix B for more details). Weestimate the log of a firm-level Cobb-Douglas production function given by:
log(y f t) = α log(kp, f t) + ω log(ki, f t) + ν log(` f t) + µ f t + ε f t, (1)
where log(y f t) is the log of output, log(kp, f t) is the log of physical capital, log(ki, f t) is thelog of intangible capital, log(`i, f t) is the log of labor, µ f t is the log of productivity and ε f t isthe error term. Estimating firm-level production functions is notoriously difficult since, forinstance, firm-level productivity µ f t is un-observable to the econometrician but it is known tothe firm at the moment of choosing its inputs. To address this endogeneity concern, we relyon three different estimation procedures proposed by the empirical industrial organizationliterature. To do so, we use the simple OLS, as a reference point. It is well known that OLSunderestimate the elasticity of quasi-fixed inputs and overestimate the elasticity of flexibleinputs due to the presence of the endogeneity problem mentioned above. Therefore, first, weuse the cost shares approach (CS) as in Foster, Haltiwanger, and Syverson (2008). Second, weuse the Olley-Pakes method (OP) by Olley and Pakes (1992) and the GMM approach from thedynamic panel data literature (DPD) as in Cooper and Haltiwanger (2006). We provide detailsregarding all methods in Appendix B.
Table 1: Production Function Estimates
(1) (2) (3) (4) (5) (6) (7) (8)
1980-1990 2000-2007
OLS CS OP DPD OLS CS OP DPDlog(y f t) log(y f t) log(y f t) log(y f t) log(y f t) log(y f t) log(y f t) log(y f t)
log(kp, f t) 0.215 0.145 0.242 0.213 0.202 0.092 0.255 0.233(0.007) (0.000) (0.011) (0.029) (0.008) (0.000) (0.039) (0.020)
log(ki, f t) 0.026 0.054 0.035 0.088 0.033 0.128 0.095 0.216(0.001) (0.000) (0.008) (0.007) (0.003) (0.000) (0.011) (0.015)
log(` f t) 0.697 0.764 0.672 0.691 0.644 0.782 0.594 0.50(0.003) (0.000) (0.012) (0.015) (0.009) (0.000) (0.011) (0.026)
Obs. 19,948 20,235 20,235 15,327 17,972 17,972 17,972 12,180
Notes: production function estimates with OLS, CS, OP and DPD. log(kp, f t) is physical capital, log(ki, f t) is intangi-ble capital, log(` f t) is the employment, and log(y f t) is value added, i.e., sales minus material costs. Material costsare not reported in Compustat since they are bundled together with labor costs in the expenditure voice Cost ofGoods Sold (see Appendix B). Therefore, to extract material costs we follow Bloom and Van Reenen (2007). Robuststandard errors are in parentheses. We control for year fixed effects for each specification. For the OLS, we alsocontrol for firm fixed effects.
Here, we estimate the production function for two different time periods (i.e. 1980-1990 and2001-2007) to highlight the rise of the input share of intangible capital. We restrict our sampleuntil 2007 to not to include the time period of the Great Financial Crises in our baseline results.In the next section, we will show the time trend in all the input shares over the whole sample
11
period to get rid of any concerns regarding the sample selection. The results of the estimationof equation 1 for the time period 1981 − 1990 and 2001 − 2007 with OLS, CS, OP and DPDare provided in Table 1. First, we notice that all the parameters are precisely estimated andqualitatively similar across different specifications. In particular, intangible capital is clearly animportant factor of production that ranges between 0.03− 0.09 in the beginning of the sample(i.e, 1981− 1990). The OLS estimation downward bias the input elasticities of both the capitalstocks.23. Moreover, it is interesting to note that the coefficient of intangible capital is muchmore down-biased than the one of physical capital. This is in line with the intuition that theinputs that are less frequently changed due to presence of adjustment friction, i.e., intangiblecapital in our case. We will return to this issue in the next section.
We take OP as our preferred estimation (i.e., ω = 0.03), since it has been the estimationtechnique most commonly used in Compustat by recent papers.24 Furthermore, we notice thatphysical capital share and the labor share are within the range of standard values.
3.1.2 Input Shares over Time
In order to document the changes in the output elasticity of intangible capital in the productionfunction, we estimate equation 1 with OLS, CS, OP and DPD between 1980-2005 using 10-yearrolling windows. We normalize the input shares in 1980 to one and document their change inthe subsequent years. Figure 1 presents the results.
Figure 1: Trends in Input shares: U.S. Compustat
1
2
3
4
Inta
ngib
le In
put s
hare
1980 1985 1990 1995 2000 2005Year
Olley-PakesCost-SharesDPDOLS
(a) Intangible Capital Input Share
.8
.9
1
1.1
1.2
Labo
r Inp
ut s
hare
1980 1985 1990 1995 2000 2005Year
(b) Labor Input Share
Note. We present output elasticities estimated with OLS (orange line - triangles), Cost shares (black line - dashed), Olley-Pakes(green line - diamonds) and Dynamic panel data (blue line - solid). We use a ten-year rolling windows to compute the inputshares over time. The standard error bands are provided in Figure B.9 in appendix.
If we look at the estimates produced by OP and DPD, our preferred measure, intangiblecapital input share today is almost thrice as much as it was in 1980. Whereas, the CS approachestimates a four log point increase in the intangible capital input share. It is evident fromthese results that the Compustat firms, which represents a significant part of the US economy,have undergone a significant change in their production technology. We label this finding asIntangible Capital Biased Technological Change, IBTC.
23This can be due to the presence of adjustment costs as explained in Olley and Pakes (1992)24Bloom and Van Reenen (2007), Imrohoroglu and Tüzel (2014) and De Loecker, Eeckhout, and Unger (2020)
12
In what follows, we notice that our estimates suggest a certain degree of substitutabilitybetween intangible capital and labor. While intangible capital share has increased the laborshare has declined in the last four decades. This is in line with the previous literature, e.g.,see Elsby, Hobijn, and Sahin (2013), Karabarbounis and Neiman (2013), and Koh, Santaeulalia-Llopis, and Zheng (2016), among others. 25
Given our estimates presented in this section, in the subsequent part of the paper we as-sume that the rise in intangible capital is an exogenous technological change in the productiontechnology that is biased towards intangible capital, IBTC. The IBTC may have been causedby the availability of advanced general purpose technologies such as ICT (i.e., Informationand Communication Technologies) or by the rising educational levels of the labor force. Un-derstanding the causes of IBTC is important but out of the scope this paper.
3.1.3 Robustness: Production Function Estimation
The robustness of the results is discussed extensively in Section B.3 in Appendix B, where weestimate the production function (baseline specification) with Ackerberg, Caves, and Frazerestimator (see Ackerberg, Caves, and Frazer 2015). This estimator allows for adjustment costsin the labor input as well (see Table B.2).
Further, differently from our baseline specification where we assume one production func-tion for the whole economy, now we assume that firms produce with same technology withinsectors (2-digit NAICS). The aggregate input elasticities are then computed as weighted aver-ages of sector-level input shares where weights are proportional to sector output. Figure B.10in appendix shows the evolution of the input shares. These results are similar to our baselineestimates, i.e., the input share of intangible capital increases and that of labor decreases overtime.
In our baseline specification, we assume Cobb-Douglas functional form for the productiontechnology. However, this choice of functional form is restrictive because it does not allowany non-linearity of or any interactions among inputs. Hence, we also consider the case of theTranslog form to address this issue. The results suggest an increasing trend in intangible inputshare (see Figure B.11 in appendix).
In our baseline measure of the intangible capital, we excluded the part of organizationalcapital. Here, we provide alternative measure of intangible capital that includes organizationalcapital as well. It has been argued in the literature that a significant part,∼ 30%, of expenditurein Selling, General and Administrative, XSGA f t in Compustat, is part of investments in organiza-tional capital. We use PIM to convert a part of XSGA f t into a stock of firm-level organizationalcapital. We document an increasing trend in the alternative measure of intangible capital (seeFigure A.1 in Appendix A). Further, we estimate the production function as described above,using the alternative measure of intangible capital. The results are provided in Table B.5 (seeAppendix B for detailed discussion). We find that the intangible capital input share has risenbetween the years 1980 and 2007, however, the absolute value may differ.
Finally, we estimate the production function with different measure of firms’ output andwith alternative set of inputs. In particular, we allow for overhead cost and material costas inputs in production and jointly estimate their input shares together with intangible and
25This result also contribute to the growing literature that speaks about the role of computer assisted technologiesin explaining the decline in the labor share. Acemoglu and Restrepo (2018) document that Automation and othercomputer-assisted technologies negatively affect labor market outcomes.
13
physical capital. All of these specifications estimate that the intangible capital is an importantpart of firms’ production function and its input share has risen significantly over time (seeSection B.3).
In the next section, we discuss the properties of intangible investment, specifically, we lookat its investment rate distribution, and compare them to those of physical investment. We usethe distribution of the investment rates to argue that intangible investment behaves differentlyfrom physical investment and therefore, an intangibles biased technological change, can haverelevant macroeconomic implications, particularly concerning firm-dynamics and market con-centration.
3.2 Fact 2: Investments in Intangible Capital are more Lumpy than Physical Capi-tal.
Having in mind that intangible capital is an important factor of production and that its impor-tance is growing over time we document the salient differences in the investment behavior offirms between intangible capital and physical capital. The investment rate for each capital isdefined as:
xj, f t
k j, f t−1≡
k j, f t − k j, f t−1
k j, f t−1+ δj, j ∈ p, i, (2)
where the depreciation of physical capital is equal to δp = 0.07, whereas the depreciationof intangible capital is equal to δi = 0.28. It is a weighted average of two different parts: 20%is the depreciation rate for the externally acquired part while we use the values mentionedin Ewens, Peters, and Wang (2019) for the knowledge capital part. Following Cooper andHaltiwanger (2006), and Clementi and Palazzo (2019), we construct a balanced panel of firmsfrom 1981-1990 to study the properties of investment rates.26 Following common practice, wealso drop observations where total value of acquisitions relative to total assets exceed 5%.27
Finally, we drop those firms that have value zero for their stock of intangible in a given year,to prevent the overestimation of the inaction rate in intangible investment.
Distribution of Investment Rates
Figure 2a and figure 2b plot the investment rate distribution for intangible and physical cap-ital, respectively. These distributions present two stark differences: First, the investment ratedistribution for intangible capital presents a clear bi-modality, with a lot of mass at the meanand around zero. Meanwhile, the investment rate distribution for physical capital is almostsymmetric around the mean, and mimics closely the findings of Clementi and Palazzo (2019).Second, the investment rate distribution for intangible capital show a small amount of negativeinvestments.28 Furthermore, the distribution of intangible capital is more fat-tailed relative tothat of physical capital.
26We also plot the investment rate distributions for the unbalanced panel in the same time period in Figure C.12in Appendix C.
27This is done to avoid biases due to acquisitions, i.e., given the accounting standards an acquisition would showup as a big investment for one firm but would not show up at all as a big disinvestment for the other. However, wenotice that in our balance panel this observations represent a small share of all entries.
28Notice that this is not by construction, i.e., is not entirely due to the capitalization of an expenditure voice likeR&D, since our measure of intangible capital indeed contains balance sheet intangibles, which, given the depreci-ation, allows for negatives.
14
Figure 2: Investment Rate Distributions: U.S. Compustat
(a) Intangible Capital (b) Physical Capital
Note. Investment rate distributions for balanced panel of firms between the years 1980 and 1990. To construct these histogramswe drop all the observation from the balanced panel - constructed in the Compustat data as explained in the text - above 2 andall the observation below -1. Results are robust to other winsorization schemes.
Table 2: Lumpiness
Investment rates Intangible Physical
Average 0.36 0.12Positive fraction, i > 1 0.90 0.90Negative fraction, i < −1 0.02 0.07
Inaction rate 0.08 0.02
Spike rate, |i| > 20 0.75 0.15Positive spikes, i > 20 0.74 0.14Negative spikes, i < −20 0.01 0.01
Standard Deviation 0.26 0.16Serial correlation, Corr(it, it−1) 0.32 0.17
We summarize the main moments of the investment rate distributions in table 2. First, no-tice that the average investment rate is much higher for intangible capital compared to physicalcapital, this is consistent with the fact that intangible capital as a share of total capital stock isincreasing over time. This also reflects a high depreciation rate for intangibles that pushesthe level of optimal investment far above those of physical capital. Moreover, as mentionedabove, intangible capital has a much higher inaction rate, defined as the fraction of investmentbelow 1% in absolute terms; particularly, intangible capital inaction rate is 8% compared to2% for physical capital. This high inactivity suggests some underlying non-convexities in theinvestment process. Furthermore, negative investment rates, defined as investment less than−1%, exhibit striking differences among the two capitals. The negative investment rate is 2%in case of intangible capital, whereas firms disinvest in physical capital around 7% of the times.Moreover, these differences are not driven by any sort of sectoral heterogeneity. We repeat thesame exercise for each sector and find similar results, see table C.8 in Appendix C. Given thefact that intangible investment is increasing over time, one might fear that the bi-modality is a
15
byproduct of different time periods and not due to non-convexities. To rule out such concerns,we provide compare moments of the intangible investment rate distributions for different timeframes in table C.9 of Appendix C. Overall, we can say that the process of intangible invest-ment is very lumpy, i.e., it entails long periods of inaction followed by booms of investmentactivity, relative to physical capital.29
This finding has particular importance with regards to how should we model intangiblecapital: dynamic or static input. In several papers such as, Korinek and Ng (2018), De Rid-der (2019), Hsieh and Rossi-Hansberg (2019), and Sandström (2020), intangible investmentsare seen as fixed costs that firms have to pay every period and it do not enter the productiontechnology directly. However, this would impose strict assumptions on the nature of intangi-ble investments. First, that the depreciation rate of intangible capital is hundred percent andthus, it is a static input. Second, intangible investments should be strongly correlated overtime as firms have to pay a cost every period to use intangibles. The data rejects both of theseassumptions. The depreciation rate for different kinds of intangible capital is higher than thatof physical capital but it is far below hundred percent. For example, average depreciation ratefor externally bought intangible is 20%. Moreover, firms do not invest in intangible capital inevery period. The serial-correlation of the intangible investment rate is 0.32. Therefore, model-ing intangible capital as a dynamic input rather than a static input or a fixed cost is importantand our framework will provide a way to analyze the properties of intangible capital, whilebeing closely related to the data.
nt towards the presence of higher capital adjustment costs for intangible capital compareto physical capital, particularly, the higher inaction rate of intangible capital suggests higherfixed costs of adjusting relative to physical capital. To confirm this intuition provided by thesesuggestive evidence we are going to calibrate the structural framework presented in section 4to match the investment rate distributions.
Next, we replicate within our dataset, being consistent with our production function esti-mation, some of the salient facts that have been documented in the literature for the US econ-omy.
3.3 Fact 3: Misallocation, Concentration, Firm size and Profit rates has increasedover time.
We focus on four main trends between 1980 and 2010: increasing firm size, increasing standarddeviation of TFPR, increasing concentration and increasing firm-level profit rates.
To calculate TFPR, we need to estimate the firm-level production function and get thecorresponding elasticities, the procedure adopted to achieve this is explained in more detailsin the next section 3.1. Once the necessary elasticity are obtained, TFPR for firm f in period tis given by:
TFPR f t = log(y f t)− α1 log(kp, f t)−ω1 log(ki, f t)− ν1 log(` f t). (3)
where y f t is firm-level output, kp, f t is physical capital, ki, f t is intangible capital, ` f t is laborand α1, ω1 and ν1 are the respective normalized output elasticities. It is important to note
29In appendix, we provide tests of normality for investment rate distributions. In particular, we compute skew-ness and kurtosis and document that both distributions are positively skewed and kurtosis is far greater than 3 (seeTable C.10). Furthermore, we also compute skewness and kurtosis for the investment rate distribution of intangiblecapital without zeros. Again, the normality is rejected as the distribution is skewed towards right and have thicktails.
16
that our definition of TFPR is in line with the one that is mentioned in Hopenhayn (2014),therefore, in the absence of capital adjustment costs, TFPR would equalize across firms. Fromequation 3, we can compute the standard deviation of the distribution of TFPR in each point intime, its normalized time series is presented in figure A.4. We find an increase in this measureof almost 50%. A similar trend has been documented in Kehrig (2015), Caggese and Perez-Orive (2017), Hsieh and Klenow (2017) and Decker, Haltiwanger, Jarmin, and Miranda (2018).However, it is important to note that our production function is different from the previousliterature, i.e., we include intangible capital together with physical capital and labor as inputsin the production technology. This measure has been interpreted in the literature as a proxyfor the allocative efficiency in the economy or misallocation, however, given the nature ofour findings, we will refer to it in the subsequent sections as standard deviation of TFPR or(mis)allocation interchangeably, as in Asker, Collard-Wexler, and De Loecker (2014).
Concentration is calculated following the spirit of Grullon et al. (2019), therefore we com-pute the Herfindahl-Hirschman Index (HHI) at SIC3 level, with the following formula:
HHIst = ∑f∈s
(y f t
∑ f∈s yst
)2
,
where s is a SIC3 sector, and y f t is output of firm f in period t. Then, we aggregate this sector-level measure of concentration using sector-level output over total output as weights. Theresulting series, as displayed in figure A.4, shows an increase of approximately 55% betweenthe year 1980 and 2005. The rise of industry concentration is a within-sector phenomenon. Asimilar trend has been documented in Autor, Dorn, Katz, Patterson, and Van Reenen (2017), forthe Census of manufacturing firms, and Grullon, Larkin, and Michaely (2019), in Compustat.
Finally, we construct our measure of firm-level profit rates following the spirit in De Loecker,Eeckhout, and Unger (2020), i.e., profit rates in our data are given by:
π f t
y f t= 1−
COGS f t
y f t−
rt(kp, f t + ki, f t)
y f t−
(XSGA f t − XRD f t)
y f t, (4)
where y f t is firm-level sales and rt is the opportunity cost of owning capital as described inAppendix A. We subtract XRD f t from XSGA f t to avoid double-counting it while calculatingprofits.30 Our aggregate measure is a yearly sales-weighted average of the firm-level measurepresented in equation 4. We find a substantial increase in the profit rate over the period ofinterest. We compare our time-series of profit rates with that of other papers in the literature,e.g., see De Loecker, Eeckhout, and Unger (2020), in figure A.5 of Appendix A. Both measuresof the profit rate has increase over the past four decades but our measure has grown less. Weimpute this variation relative to the previous literature to the difference in the definition ofprofit rates used in this paper (i.e., we account for the cost of holding intangible capital).31
Finally, we compute average firm size in the US Compustat (see Figure A.6 in appendix).We find that firm size has increased in the last three decades (both in terms of sales and em-
30We know, as explained also in Peters and Taylor (2017), that most of the firms that report a positive expenditurein Research and Development include also this expenditure in Selling, General and Administrative. If we wouldnot subtract XRD f t from XSGA f t we would count R&D both as capital and as expenditure.
31This is particularly important given the fact that intangibles have increased from 10 percent to almost 50 percentof the total capital stock, therefore, we argue that this omission of the cost of holding intangibles in the earlierspecification of the profit rate led to its rise.
17
ployment). We also confirm this finding by plotting the average firms size in the BDS datasetthat encompasses the whole US economy (see Figure A.7 in appendix).
4 Theoretical Framework
In this section, we present a structural model of investment dynamics with two dynamic in-puts (physical and intangible capital) and endogenous firm entry and exit to connect all threestylized facts together that are presented in the previous section. The model includes a richand flexible structure of investment adjustment costs for both capitals that will help us to im-itate the investment rate distribution of both capitals as presented in Section 3.2 as Fact 2.The presence of capital adjustment costs in this environment creates a dispersion in ARPKi &ARPKp and generate (mis)allocation. Finally, we will use this framework to highlight the linkbetween the share of intangible input in production, the overall dispersion in TFPR, firm sizeand industry concentration and connect the Fact 1 with Fact 3 (see Section 3.1 & Section 3.3).
4.1 Model Environment
The model follows the spirit of Clementi and Palazzo (2016b). Time is discrete and indexedby t = 1, 2, . . . . At time t a positive mass of price-taking firms produce an homogeneousgood by means of the production function y = ezkα
pkωi `
ν, with α, ω, ν in (0, 1). Where kp wedenote physical capital, ki is intangible capital, l is labor and z is the idiosyncratic randomproductivity.
Idiosyncratic productivity z is driven by the stochastic process:
z′ = ρzz + σzε′,
where ε ∼ N (0, 1). The conditional distribution of z will be denoted by Γ(z′|z).Firms discount future profits by means of the time-invariant discount factor 1
R , R > 1.Physical capital depreciates at a rate δp ∈ (0, 1), whereas, intangible capital depreciates at arate δi ∈ (0, 1). Adjusting physical capital stock by xp and intangible capital stock by xi bearthe cost:
C(xp, xi; kp, ki) =γp
2
(xp
kp
)2
kp +γi
2
(xi
ki
)2
ki + 1xp 6= 0 fpkp + 1xi 6= 0 fiki,
where γp, γi, fp, fi ∈ R+. We allow for two different kinds of adjustment costs: convex andfixed. We do not allow for irreversibilities in investment in the baseline version of the model.32
Generally, these non-convex costs of adjustment are intended to capture indivisibilities in cap-ital, increasing returns in the installation of new capital, and increasing returns to retrainingand restructuring of production activity. Moreover, this formulation of non-convex adjustmentcosts can be interpreted as a mild form of irreversibility, as disinvestment bears a cost in termsof output, which stems for the potential specific nature of capital. Specifically if capital is tai-
32Irreversibility is defined as the difference between the price of buying new capital and the price of sellingused capital, see Cooper and Haltiwanger (2006).We do not include them since for physical capital, as mentionedin Clementi and Palazzo (2019), the distribution of investment rates is symmetric. Meanwhile, irreversibility inintangible capital is not allowed due to the presence of issues related to their identification, see detailed discussionin section 5
18
Figure 3: Timing in the Model
time t
Incumbents observeproductivity shock
HireLabor
Pay c f &Produce
xp > 0, xi > 0
xp > 0, xi = 0
xp = 0, xi > 0
Wait
Exit
time t + 1
Potential Entrantsobserve Productivity
Do not enter
pay ce Invests
lored to some particular needs of a firm it can in principle be very difficult to resell it.33 Theconvex costs capture overtime costs, inventory costs, and machine set-up costs. Furthermore,we assume that the capital adjustment costs are proportional to their respective capital stock:this is a common specification that is used in order to take care of the size effect. Finally, as itwill become clear later, in our baseline specification the adjustment costs are paid in terms offinal output.
We assume that the demand for firm’s output and the supply of both capitals are infinitelyelastic and we normalize their prices at 1. The supply of labor is given by L(w) = wψ, whereψ > 0 and w ∈ R+ is the real wage.
Operating firms incur each period a fixed cost c f > 0; this is usually interpreted as a per-period expense that firms must incur in order to operate, e.g., to hire one unit of managerialactivity. Firms that quit production cannot re-enter the market at a later stage and recoup theundepreciated part of their capital stocks, net of the adjustment cost.
Every period there is a constant exogenous mass m > 0 of prospective entrants, each ofwhich receives an initial productivity s, with s ∼ Λ(s), a Pareto distribution with scale param-eter η. Conditional on entry, the distribution of the idiosyncratic shock in the first period ofoperation is Γ(z′|s), strictly increasing in s. Entrepreneurs that decide to enter the industry payan entry cost ce ≥ 0.
Finally, in each period, the stationary distribution of operating firms over the three dimen-sions of heterogeneity is denoted by Ω(z, kp, ki; w). A comprehensive picture of timing in themodel is presented in Figure 3.
33This idea that intangible capital is very specific to the needs of the firm that uses it has been suggested by Haskeland Westlake (2018) and Edmond, Midrigan, and Xu (2018). However, we move here a step forward compared tothem and we test this hypothesis concretely specifying a flexible model to be tested in the data - in principle ourmodel could reject this estimating fi to be close to zero.
19
4.2 The incumbent’s problem
Given idiosyncratic productivity z, physical capital kp and intangible capital ki, the profits ofan incumbent are given by:
π(z, kp, ki; w) = max`
ezkαpkω
i `ν − w`. (5)
Upon exit, a firm obtains a value equal to the undepreciated portion of its physical capitalkp and intangible capital ki, net of the adjustment cost it incurs in order to dismantle it, i.e.:
Vx(kp, ki) = (1− δp)kp + (1− δi)ki − C(−(1− δp)kp,−(1− δi)ki; kp, ki).
Then, the start-of-period value of an incumbent firm is dictated by the function V(z, kp, ki; w)
which solves the following functional equation:
V(z, kp, ki; w) = π(z, kp, ki; w)
+ maxVx(kp, ki), V1(z, kp, ki; w)− c f ,
V2(z, kp, ki; w)− c f , V3(z, kp, ki; w)− c f , V4(z, kp, ki; w)− c f ;(6)
where the value of investing in both capital is given by:
V1(z, kp, ki; w) = maxk′p,k′i−xp − xi − C(xp, xi; kp, ki) +
1R
∫V(z′, k′p, k′i; w)Γ(dz′|z),
s.t. k′p = (1− δp)kp + xp,
k′i = (1− δi)ki + xi;
(7)
the value of investing in only physical capital is given by:
V2(z, kp, ki; w) = maxk′p−xp − C(xp, 0; kp, ki) +
1R
∫V(z′, k′p, (1− δi)ki; w)Γ(dz′|z),
s.t. k′p = (1− δp)kp + xp;(8)
the value of investing in only intangible capital is instead given by:
V3(z, kp, ki; w) = maxk′i−xi − C(0, xi; kp, ki) +
1R
∫V(z′, (1− δp)kp, k′i; w)Γ(dz′|z),
s.t. k′i = (1− δi)ki + xi;(9)
finally, the value of waiting is given by:
V4(z, kp, ki; w) =1R
∫V(z′, (1− δp)kp, (1− δi)ki; w)Γ(dz′|z). (10)
4.2.1 The entrant’s problem
The value of a potential entrant that draws an initial productivity s, where s ∼ Λ(s), is givenby:
Ve(s; w) = maxk′p,k′i−k′p − k′i +
1R
∫V(z′, k′p, k′i; w)Γ(dz′|s); (11)
20
the potential entrant will invest and start operating if and only if Ve(s) ≥ ce.
4.2.2 Recursive Competitive Equilibrium
The Recursive Competitive Equilibrium (RCE) consists of (i) value functions V(z, kp, ki; w),V1(z, kp, ki; w), V2(z, kp, ki; w), V3(z, kp, ki; w), V4(z, kp, ki; w) and Ve(s; w), (ii) policy functions`(z, kp, ki; w), xp(z, kp, ki; w), xi(z, kp, ki; w), k′p(s; w), k′i(s; w), and (iii) an incumbents’ measureΩ(z, kp, ki; w), and an entrants’ measure E(z, kp, ki; w) such that:
1. V(z, kp, ki; w), V1(z, kp, ki; w), V2(z, kp, ki; w), V3(z, kp, ki; w), V4(z, kp, ki; w), `(z, kp, ki; w),xp(z, kp, ki; w) and xi(z, kp, ki; w) solve (5), (6), (7), (8), (9), and (10);
2. Ve(s, s), k′p(s, s) and k′i(s, s) solve (11);
3. The labor market clears:∫`(z, kp, ki; w)dµ(z, kp, ki; w) = L(w)
4. For all Borel sets Z ×Kp ×Ki ⊂ R+ × R+ × R+,
E(Z ×Kp ×Ki; w) = m∫Z
∫Be(Kp,Ki ;w)
Λ(ds)Γ(dz′|s),
where Be(Kp,Ki; w) =
z s.t. k′p(s; w) ∈ Kp, k′i(s; w) ∈ Ki and Ve(s; w) ≥ ce
;
5. For all Borel sets Z ×Kp ×Ki ⊂ R+ × R+ × R+ and ∀t ≥ 0,
Ω(Z ×Kp ×Ki; w) =∫Z
∫B(Kp,Ki ;w)
Ω(dz, dkp, dki; w)Γ(dz′|z) + E(Z ×Kp ×Ki; w),
where B(Kp,Ki, s) =(z, kp, ki) s.t. maxV1(z, kp, ki; w), V2(z, kp, ki; w), V3(z, kp, ki; w),
V4(z, kp, ki; w) − c f ≥ Vx(kp, ki), (1 − δp)kp + xp(z, kp, ki; w) ∈ Kp and (1 − δi)ki +
xi(z, kp, ki; w) ∈ Ki
.
4.3 Discussion: Adjustment Costs and Firm Investment
Here, we discuss the extensive and the intensive margins of firm investments and study itsimplications for factor returns and TFPR. In particular, we focus on the intangible capitalinvestments, however, all of these results apply directly to the physical capital investments aswell.
The extensive margin of firm’s intangible investment is characterized by the inaction re-gion [ki(z, kp, ki), ki(z, kp, ki)]. At any time period t, firm observe their productivity shock andchoose to act only if its current capital stock is far away from the optimal level (i.e., below kor above k). If the current stock of capital is too small (below k), firms choose to invest and ifthe current stock is too large (above k), it decides to disinvest. Further, an increase in the fixedcost widens the inaction region. In order to highlight this mechanism, let us focus on the valueof only investing in intangible capital V3(z, kp, ki; w) and the value of waiting V4(z, kp, ki; w).Firms invest if and only if V3(z, kp, ki; w) ≥ V4(z, kp, ki; w).
It is clear from the characterization of the value function in Section 4 that firm’s decisionto invest today depends on its impact on the continuation value (expected discounted sum offuture cash-flows). By using Equations 9 & 10, and rearranging terms, it can show that firmsinvest only if the expected future gains from investment are higher than the costs that firms
21
have to pay today (see Equation 12). The left hand side (LHS) of the equation represents thebenefit of investing. It is equal to zero if firm’s stock of capital is at the optimum level fora given productivity shock k∗ = k
′i = (1− δi)ki, and in this case the cost of action is fi ki.
Therefore, inaction is relatively more profitable. Moreover, the LHS is decreasing and convexfunction in ki.34 The cost of action, for reasonable parameterization, is also decreasing in ki,intersecting the LHS at ki and ki.
E[V(z′, (1− δp)kp, k′i)−V(z′, (1− δp)kp, (1− δi)ki)
]︸ ︷︷ ︸Benefit of action
≥ fi ki +γi
2
(xi
ki
)2
ki + xi︸ ︷︷ ︸Cost of action
. (12)
In Figure 4a, we plot the inaction rate against the parameter that discipline the fixed ad-justment cost fi. The model predicts a monotonically increasing relation between inaction rateand fi. The reason being that the inaction region [ki, ki] outside which firms act, increases infi. We exploit this information to identify the parameters of fixed adjustment costs in the nextsection.
The intensive Margin of firm’s intangible investment is characterized by its marginal returns.In this economy, due to the Cobb-Douglas assumption, marginal returns are proportional toaverage returns. The first order condition that pin downs the optimal investment in intangiblecapital at the intensive margin is described by,
E
[ωeARPK
′i −
∂C ′i∂k′i
]= r + δi + R
∂Ci
∂k′i, (13)
where, ARPKi = log(yi) − log(ki) is the average revenue product of intangible capital. Ci
represent adjustment costs associated with intangible capital at time t. In the absence of ad-justment costs, the firm would respond to productivity shock by investing enough such thatthe expected value of marginal product of intangible capital, E ωeARPK
′i , is equalized to the
marginal cost of capital (r + δi). This makes investment serially uncorrelated over time. How-ever, in the presence of convex adjustment costs firms builds its capital stock slowly over time.This effect is captured by Figure 4b, where we plot corr(xi, x′i) against the parameter of con-vex adjustment cost γi. The model predicts that the serial-correlation of investment rate ismonotonically increasing, whereas, the inaction rate remains unchanged.
In this model, the dispersion in factor returns is caused by two forces: the uncertainty inthe productivity shock (risk in capital accumulation), and the presence of adjustment costs.The effect of adjustment costs on firm investment behavior are captured by the derivatives ofthe adjustment cost function C and C ′ , where the former represents the cost of changing capitalstock today and the later outlines the cost of re-changing the capital stock in the next periodby one unit (due to uncertainty in productivity). We show in Figure 5 that the dispersion inARPKi is increasing in γi (the importance of convex adjustment cost due to the intangibleinvestments), however, ARPKp does not change.
The dispersion in TFPR in our framework is a weighted geometric mean of the average prod-uct of inputs, where the weights are proportional to their output elasticities (see Hopenhayn
34The Inada conditions ensure that the slope is very negative at lower levels of capital stock and it flattens out ascapital stock increases.
22
Figure 4: Adjustment Costs and Investment Rate: Sensitivity Analysis
7
8
9
10
11
12
Inac
tion
Rate
(%)
.3
.32
.34
.36
.38
.4
corr
(xi,t
,xi,t
+1)
.002 .003 .004 .005 .006Fixed Cost - Intangible
corr(xi,t,xi,t+1)Inaction Rate (%)
(a) Extensive Margin
7
8
9
10
11
12
Inac
tion
Rate
(%)
.3
.32
.34
.36
.38
.4
corr
(xi,t
,xi,t
+1)
.05 .1 .15 .2 .25Convex Cost - Intangible
(b) Intensive Margin
Note. Model computations.
Figure 5: Dispersion in ARPKp and ARPKi
.55
.6
.65
.7
.75
Var
(ARP
K p)
3
3.5
4
4.5
5
5.5
Var
(ARP
K i)
.05 .07 .09 .11 .13Convex Cost - Intangible
Var(ARPKi) Var(ARPKp)
Note. Model Computations for the dispersion in ARPKp and ARPKi against the parameter of convex adjustment cost γi .
(2014) for detailed discussion). As presented in Equation 14, it is evident that TFPR is notequalized across-firms. In particular, the difference emerge due to the dispersion in the factorreturns (ARPKi and ARPKp).
TFPR = ν1 (ARPL) + ω1 (ARPKi) + α1(
ARPKp)
(14)
In particular, our measure of misallocation in the economy is defined by,
Var(TFPR) = ω12Var (ARPKi) + α1
2Var(
ARPKp)+ 2α1ω1Cov(ARPKi, ARPKp) (15)
where Var represents variance in the variable and Cov is the covariance. The misallocationin this economy is independent of ARPL as it is equalized across firms. The misallocation isdriven by the dispersion in ARPKi and ARPKp. The importance of the dispersion in factor
23
returns crucially depends on their input share.35 Therefore, as discussed in Section 6, IBTC,where intangible input share increases and labor input share declines, matters for observedmisallocation in the economy.
5 Structural Analysis
In this section, we use the structural framework presented in section 4 to estimate the ad-justment costs associated with physical and intangible capital and to evaluate the investmentbehavior as we observe in the data. In order to do so, we use a set of identifying moments, aspresented in the previous sections, to discipline the un-observable parameters of the model.Then, we validate our model on non-targeted moments of the investment rate distributionsand further discuss the implications of the presence of adjustment costs on the average prod-uct of inputs.
5.1 Data for Empirical Moments
In order to precisely estimate the model parameters, we employ data from various sources. Inparticular, first, we use Compustat to compute the moments of investment rate distribution forboth capitals as described in Section 3.2. However, as discussed earlier, Compustat containslarge firms which makes it unsuitable to compute moments regrading firm life-cycle. As aresult, we move to BDS data that encompass the universe of firms in US. Here, we computefirm entry and exit rate, and average firm and entrant size.
5.2 Identification
The baseline calibration matches jointly the investment behavior of physical and intangiblecapital at the micro level and business dynamism in the overall US economy for the sampleperiod 1980-1990. The parameterization proceeds in two steps. First, we fix a set of parametersthat are estimated outside the model, e.g., the parameters governing the production technol-ogy and the TFP process. Second, given the values of these fixed parameters, we choose theremaining parameters to match informative moments regarding firms’ investment distributionand firms’ life-cycle.
Fixed parameters A model period is one year, so we set the interest rate R = 1.05. Theannual depreciation rate for physical capital is δp = 0.07, which is equal to the value used toperform the above empirical analysis. We set the depreciation rate for intangible capital to be,δi = 0.28, using an average of the measures provided by Ewens, Peters, and Wang (2019) asextensively discussed in section 3.2. We set the measure of potential entrants to m = 0.03: thisis inconsequential, as it only influences the number of firms in the steady state and not themoments of interest. The production function parameters are estimated using Olley and Pakes(1992) method as reported in table 1. The results are robust to all remaining three methods asmentioned in section 3.1. Finally, the persistence of the idiosyncratic process is ρz = 0.85 andthe standard deviation is σz = 0.20 (see Foster, Haltiwanger, and Syverson, 2008 and Clementiand Palazzo, 2016a).36
35In Equation 15, α1 = αα+ν+ω .
36We do not use the production function estimation to estimate the values of TFP process because our samplecontains only large firms and thus, it is unsuitable to use that process to match the overall cross-section of firms in
24
Fitted parameters We choose the remaining parameters to match some of the momentspresented in table 2. Specifically, we use inaction rates, i.e., investment rates that are within±1%, to discipline the parameters governing the fixed costs in both physical and intangiblecapital, i.e., fp and fi. This is particularly appealing since the model predicts that the fixed costsof adjusting directly influence the extensive margin of investment, i.e., the amount of actionand inaction in investment of a given capital. Meanwhile, we use the serial correlation of bothinvestment rates to identify the convex costs of adjusting for both capitals, i.e., γp and γi. Highconvex costs makes the firms to adjust their capital stock more slowly over time, which inturn increases the autocorrelation of investment at the firm level. Finally, to identify the entrycost ce, the operating cost c f and η, the parameter that governs the Pareto distribution of theproductivity of potential entrants, we match the entry rate, the average size of incumbents andthe average size of entrants.37
The parameters are estimated using the following routine. For arbitrary values of the vectorof parameters, Θ = (γp, γi, fp, fi, ce, c f , η), the dynamic programming problem is solved andthe policy functions for investment in both capitals, for entry and for exit are generated. Usingthese policy functions, the decision rules are simulated until the distribution of firms overz, kp, ki is converged. We draw from this stationary distribution simulating the economy for20 period and construct a balanced panel of firms in the same spirit of the empirical analysispresented above. We compute the entry rate, the average size of entrants and the averagesize of exiters from the stationary distribution and targeted moments of investment rates fromthe simulated panel, which we denote as Ψ(Θ). We estimate the fitted parameters Θ using aminimum distance criterion given by:
L(Θ) = minΘ
(Ψ−Ψ(Θ))′W(Ψ−Ψ(Θ)). (16)
Following Asker, Collard-Wexler, and De Loecker (2014), we set the weighting matrix W =
I and use grid search to find the vector Θ that minimize the objective function.
5.3 Results
The fitted parameters from the simulated method of moments and the implied moments ofthe model are presented in table 3. The model identifies very different adjustment costs forphysical and intangible capital. Similar to Clementi and Palazzo (2019), our model imputesalmost negligible fixed costs and low convex costs of adjustment to physical capital. The rea-son being that the Compustat dataset contains disproportionately large firms.38 Moreover, ourmodel implies that intangible capital entails much higher adjustment costs, particularly thefixed costs relative to physical capital. The fixed costs in this model captures the indivisibili-ties and increasing returns to restructuring, among others and these are particularly large forintangible capital. The estimation also implies higher convex costs for intangible capital rela-tive to physical capital. For instance, consider the case of process innovation (e.g., investmentsin the implementation of Just In Time manufacturing techniques to streamline production pro-
the economy (see Appendix D.1 for detailed discussion).37We assume that potential entrants draw their first productivity s from Λ(s), a Pareto distribution.38However, contrary to our results, Cooper and Haltiwanger (2006) use a more heterogeneous sample of plants
from the confidential Census database and find larger adjustment costs. Another point of distinction is that ouranalysis is at firm level. Therefore, we want to emphasize that our estimates can be interpreted as a lower boundsof these costs.
25
Table 3: Estimated parameters and Moments
Fixed Value Description
R 1.05 Annual interest rateδp 0.07 Annual depreciation rate physical capitalδi 0.28 Annual depreciation rate intangible capitalm 0.03 Measure of prospective entrantsα 0.24 Physical capital shareω 0.03 Intangible capital shareν 0.67 Labor shareρz 0.85 Autocorrelation idiosyncratic productivityσz 0.20 Standard deviation idiosyncratic productivity
Fitted Value Description Moments Model Data
γp 0.037 Convex adj. cost physical capital corr(xp, f t, xp, f t−1) 0.33 0.32γi 0.165 Convex adj. cost Intangibles corr(xi, f t, xi, f t−1) 0.17 0.17fp 7e-5 Fixed adj. cost physical capital Inaction rate: xp 0.01 0.02fi 4.4e-3 Fixed adj. cost Intangibles Inaction rate: xi 0.08 0.08ce 0.11 Entry cost Entry Rate 0.12 0.13c f 2.05 Operating cost Avg. firm size 20.6 20.6η 3.05 Scale parameter Avg. Entrant size 5.5 5.6
cess). As discussed before, this requires involvement of all the departments within a firm andcannot be done at a more disintegrated level. Furthermore, it needs long set-up times, businessrestructuring and workers’ retraining that makes this process very costly and cumbersome.39
The model performs well on non-targeted moments such as mean investment rate, positiveand negative investment rate, and positive spike rate. We list them in table D.12 in AppendixD. In line with data, the model identifies higher spikes in the investment rate distribution rela-tive to that of physical capital. The bi-modality of the investment rate distribution of intangiblecapital is a natural byproduct of the model. It stands for the fact that in the presence of highfixed costs and depreciation rates, either firms invest a lot in intangibles, or they do not investat all and so they remain inactive. Meanwhile, the distribution of investment rate for physi-cal capital is close to normal, reflecting the underling distribution of firms’ productivity, as inthe data. As a final remark, we want to emphasize, as already mentioned in section 4, thatour model performs reasonably well without requiring any sort of irreversibility in intangi-ble capital (i.e., negative investment rate is much lower for intangible capital relative to that ofphysical capital). This is due to the fact that with such a high depreciation rate and with a hugefixed cost of adjusting, when a negative productivity shocks hits a firms they do not dis-investimmediately, as they prefer to wait, letting the capital to depreciate, instead of paying the fixed
39Another good example is the insurance business in the healthcare sector, the most intangible-intensive sectorin the US - both in Compustat and BEA - that largely revolves around information management and therefore,requires investment in large scale computer system, e.g., see Cutler and Garber 2004. This entails large fixed costsbut the system can process information of one more individual at almost zero cost - as long as there are no capacityconstraints. This is part of what we capture with high fixed costs of adjustment in the model.
26
cost.Here, we discuss the cross-sectional and lifecycle implications of the model. Similar to
what is documented in the previous literature on firm dynamics, our model exhibits the sizeand age distributions that are right-skewed (see Figure 6). The model further predicts thatsmaller firms exhibit faster growth rates. Overall, the model does a good job in matching thefirm size distribution that is present in the data (see Figure 6a). The majority of firms in themodel are small, whereas, a large portion of employment is concentrated among the largefirms, a feature well established in the data, (see Figure 6b). Finally, the model predicts thatthe around 40% of the firms are operating for more than 11 years and they account for around60% of the employment share, which is close to the figures we find in the data, (see Figure 6cfor cohort-wise employment shares and Figure 6d for age distribution).
Figure 6: Size and Age Distribution
upto 4 (4-9] (9-19] (19-49] 49+
Employment Bins
0
20
40
60
Em
plo
ym
en
t S
ha
re (
%)
Model
BDS Data
(a) Employment Distribution
upto 4 (4-9] (9-19] (19-49] 49+
Employment Bins
0
10
20
30
40
50
60
Firm
s (
%)
(b) Firm Distributions
1 2 3-5 6-10 11+
Age Bins
0
20
40
60
Em
plo
ym
en
t S
ha
re (
%)
(c) Cohort Employment share
1 2 3-5 6-10 11+
Age Bins
0
10
20
30
40
50
Firm
s (
%)
(d) Age distribution
Note. We use the BDS data for empirical moments.
5.3.1 Quasi-Fixed Inputs and Marginal Products
Here, we discuss the consequences of high adjustment costs on firms’ response to productivityshocks, particularly we focus on the volatility and the dispersion in the marginal product ofboth capitals. The fact that capital is quasi-fixed, i.e., it entails adjustment costs on top of time
27
to build, leads to a situation where the marginal product of each capital is not equalized acrossfirms. This dispersion in marginal product is interpreted as the misallocaiton of resources.This happens because, when a productivity shocks hits, the firm cannot adjusts the capitalstock immediately to the optimal frictionless level; therefore, the marginal product of capitaldeviates from the marginal cost, i.e., the opportunity cost of holding the capital.40 Given thatour estimation pointed out that intangible capital is more fixed compared to physical capital,our model predicts that the marginal product of intangible capital is more dispersed as well.
Here, we provide the empirical evidence for our model prediction. However, measuringmarginal product of capital in the data is not straightforward. Therefore, we compute averageproduct of each capital and interpret it as marginal product under our assumption of Cobb-Douglas production technology (i.e., marginal product of each capital is equal to their averageproduct times their respective output elasticity). Furthermore, under the assumption that allfirms use the same production technology within a sector, the dispersion in marginal productof each capital is directly translated into the dispersion in their respective average products.To do this end, we compute in the data the log of the average product of both capitals, for firmf at time t as:
ARPKj, f t = log(y f t)− log(k j, f t), j ∈ p, i, (17)
where y f t is firm-level output and k j, f t is firm-level capital. We compute the dispersion in theaverage product of capital at the SIC2 level. Results are presented in figure 7, where we scatterplot the standard deviation in ARPKi against that of ARPKp for each sector. It is evident fromthis figure that in every sector the average product of intangible capital is more dispersed thanthat of physical capital (see Figure D.13 in appendix for a comparison of distribution of averageproducts of both capitals between data and model). We further plot the dispersion in ARPKi
against the dispersion in ARPKp for each industry at SIC3 level and document that sd(ARPKi)
is higher than sd(ARPKp) in almost all the industries (see Figure D.14 in Appendix D.2).Furthermore, we provide additional evidence to corroborate our findings on the dispersion
in average products. particular, it is possible that part of the dispersion in the average productof intangible capital as computed in the data is driven by measurement error. To overcomethese concerns, we run the following regression specification:
ARPKj, f t = γ0 + γ1ε f t + γ2k j, f t + γ3z f t−1 + γs + γt + ε f t, j ∈ p, i, (18)
where the dependent variable ARPKj, f t is the average product of input k j and ε it equals z f t −z f t−1, which we interpret from the lens of the model as a structural exogenous innovation tofirm productivity. We control for firm-level input k j at time t and lagged productivity z f t−1 tosee how the average products of two comparable firms change when they are hit by heteroge-neous productivity shocks. Finally, γs and γt are sector and year fixed effects, respectively, andγ1, which is the coefficient of interest, captures how average product respond to a productivityshock.
Results are presented in table 4. First, notice that in a static model with no adjustment costs
40The dispersion in the marginal product of capital has been interpreted as misallocation in a recent literaturestarted by Hsieh and Klenow (2009). More recently, Bartelsman, Haltiwanger, and Scarpetta (2013) and Asker,Collard-Wexler, and De Loecker (2014) argued that dynamics inputs that entail adjustment costs have more dis-persed marginal product compared to static inputs like labor, casting doubts on the interpretation of such a disper-sion as misallocation of resources.
28
Figure 7: Sectoral Dispersion in ARPKp and ARPKi
0
.4
.8
1.2
1.6
2
2.4
sd(A
RPK p
)
0 .4 .8 1.2 1.6 2 2.4sd(ARPKi)
SIC2-Sectors
Note. It plots the standard deviation of ARPKp on the y-axis and standard deviation of ARPKi , on the x-axis, averaged over allyears in the sample period. Average products are measured using equation 17. Each circle represents a SIC2 sector, where size ofthe circle is proportional to its size (sale-weighted) in Compustat.
Table 4: Heterogeneous Response of Average products to TFPR shocks
(1) (2)
Dependent Variable ARPKp, f t ARPKi, f t
ε it 1.306 1.559(0.022) (0.046)
kp, f t -0.141(0.002)
TFPR f t−1 1.255 1.410(0.011) (0.023)
ki, f t -0.452(0.003)
Time dummies X XSector dummies X XObservations 17,543 17,543R-squared 0.697 0.661
Notes. We report the coefficient of the regression ARPKp, f t and ARPKi, f t against productivity shock εit. Thecontrols include TFPR f t−1, lagged productivity, kp, f t, physical capital, and ki, f t, intangible capital. We restrict thesample period between 1980-1990. See Table D.13 for regression results for the full sample. Standard errors are inthe parentheses. All coefficients re significant at 1% level.
and time to build, the value of γ1 should be zero, as the average product does not respondto productivity shocks. However, as predicted by the model, we find that the average prod-ucts of physical and intangible capital are reactive to productivity shocks, with γ1 significantlyabove zero. Moreover, the average product of intangible capital is more reactive to productiv-ity shocks relative to the average product of physical capital. This makes the average productof intangible capital more volatile. This is in line with the predictions of the model that firms
29
do not adjust their intangible capital frequently due to the presence of high adjustment costs,as already presented in figure 7. Furthermore, by using the within firm variation in the produc-tivity shocks, we show that the average product of intangible capital reacts more to the shocksthan the average product of physical capital at the firm level (see Table D.14 in Appendix D.2).
6 IBTC and the Macroeconomic Implications
Until now, we have shown that intangible capital is an important part of the firms’ productionprocess and its input share in production is rising over time at the expense of labor, a phe-nomenon we label as IBTC. Further, we show that intangible capital entails higher adjustmentcosts relative to physical capital due to the presence of inherent non-convexities in the invest-ment process. In this section, we study the macroeconomic implication of IBTC. In particular,we examine how the economy responds when firms’ production function undergoes a shiftfrom a flexible input, labor, to a quasi-fixed input, intangible capital, and further we assessthe quantitative importance of IBTC in explaining some of the major trends documented forthe US economy. Particularly, we focus our analysis on empirical trends that have recently re-ceived a lot of attention in the literature, such as concentration as measured by HHI, dispersionin TFPR, sales-weighted profit rates, intangible investment share, and average firm size.
To do so, we take a two step approach. In a first step, we solve the model for different lev-els of intangible capital input share in production. In particular, we increase the value of theintangible capital input share, ω and decrease the labor input share in production, ν, such thatreturns to scale remains constant. We discuss the model predictions and highlight the mainmechanism. Further, we exploit the cross-sectoral variation in intangible intensity to provideempirical evidence that corroborates the main model predictions. Once we establish the valid-ity of our main results, in the second step, we compare the model outcomes in two differentsteady states between the years 1980 and 2007. First, the 1980 steady state is represented by ourbenchmark calibration (i.e., average economy between the years 1980 and 1990) as discussedin Section 5, whereas we compute 2007 steady state with firm-level production function es-timated between 2001-2007 (see Table 1 for production function estimates), while keeping allother parameters constant (i.e., their values remain same as in the 1980 steady state). This willhelp us to highlight the macroeconomic effects of that are strictly caused by IBTC. Throughthis final exercise, we quantify the change predicted by the model between these two steadystates in the concentration as measured by HHI, average firm size, average entrant size, intan-gible investment share and the dispersion in TFPR and compare them to what we find in thedata.
6.1 Intangible Intensity and Firm Dynamics
Due to the fact that intangible capital is a quasi-fixed input, while labor is a flexible input inproduction, the various moments that define firm dynamics may significantly differ acrossthe economies that are heterogeneous in their usage of intangible capital relative to labor. Inthis section, using our quantitative framework, we examine how changes in intangible capitalinput share relative to labor input share affects various economy-level variables such as equi-librium wages, overall employment, number of operating firms, intangible intensity (i.e., thestock of intangible capital over employment), intangible investment over output, distribution
30
of TFPR, sales-weighted profit rates, concentration as measured by HHI and average firm size.In particular, we solve the model several times to replicate various sectors that are hetero-
geneous in their usage of intangible capital (i.e., the ratio of intangible capital input share, ω,to labor input share in production, ν), keeping fixed the returns to scale. We compute all mo-ments of interest and report them in Figure 8 (see Table D.15 in appendix for correspondingvalues). We normalize all variables to their values in the benchmark economy. First, note thatthe average intangible intensity in the economy increases monotonically with increasing valueof ω/ν. This allows us to use intangible intensity, ki/l, as an empirical counterpart for ω/ν,and therefore, we use them interchangeably from here onwards. Second, the model predictsthat the average intangible investment over sales, dispersion in TFPR, sales-weighted profitrates, average firm size and concentration are increasing in intangible intensity.
The intuition behind these results is straightforward: as the sector becomes more intangibleintensive, firms needs less labor and more intangible capital to produce the same amount ofgoods, therefore, by construction the labor share declines and the intangible investment shareincreases. Furthermore, increasing the intangible intensity of a sector means de facto that weare moving the firms from using an input that is relatively cheap to adjust, labor, to usingan input that is costly to adjust, intangible capital, reducing therefore the economic profits offirms and the value of entry. This means that the productivity threshold of the marginal entrantincreases, which further implies that only very productive firms are able to operate profitablyin this environment, consequently increasing the average productivity and as well as averagesize (i.e., number of employees) of incumbent firms in intangible-intensive environment. Inother words, the selection effect is stronger in intangible-intensive sectors.
Furthermore, due to the nature of adjustment costs, we document a reallocation of eco-nomic activity from small firms towards large firms. This happens because, a rise in adjust-ments costs (fueled by the rise in the intangible input share), makes it more difficult for smallfirms to grow – particularly new entrants. However, large (high productivity) firms, as theyexpect lower productivity in the future, due to a mean reverting productivity process, have anoutside option of letting the capital stock to depreciate without incurring any additional cost ofadjusting. A decline in the mass of firms and size-biased effect of the adjustment costs implieshigher concentration in intangible-intensive sectors as measured by the HHI index. Moreover,a decline in the mass of firms leads to lower labor demand and as a results, the equilibriumwage declines.
Meanwhile, we observe an increase in sale-weighted profit rates in the model. This is a by-product of two forces: the increase in average productivity of incumbents (improved selection)and the decline in equilibrium wages. Finally, the model predicts that intangible-intensivesectors have high dispersion in TFPR. This happens because TFPR in our framework is aweighted geometric mean of average product of inputs, where the weights are proportional totheir output elasticities. Furthermore, it is important to note that in the absence of adjustmentcosts (including time-to-build), TFPR would equalize across firms. However, due to the pres-ence of capital adjustment costs, average product of each capital is not equalized and that inturn shows up as the dispersion in TFPR. Therefore, when we increase the intangible intensityof a sector in our setting, firms’ investment as a share of their output are moving from a flexibleinput, labor which is characterized by zero dispersion in its average product in the model, to arelatively fixed one, intangible capital which is characterized by large dispersion in its averageproduct, the dispersion in TFPR increases as well.
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Figure 8: Intangible Share and Firm-Dynamics
In what follows, we provide empirical evidence for the main predictions discussed above.In doing so, we use U.S. Compustat data between 1980 and 1990. We define sectors at SIC2level, and we compute intangible intensity for each sector, along with other variables of in-terest such as concentration, dispersion in TFPR, sales-weighted profit rates and intangibleinvestment over output. The measurement methodology of these variables is explained in sec-tion 2.1. We present the scatter plots for these variables against intangible intensity in Figure 9(see Table D.16 in appendix for corresponding regressions with time and sector fixed effects).
Similar to what is predicted by the model, we find that firms in intangible-intensive sectorsspend more in intangible investments as a percentage of their output (see Figure 9a). Further,the model predictions about increasing (mis)allocation, i.e., increasing dispersion in TFPR, insectors’ intangible intensity is also supported in the data (see Figure 9b). This finding suggeststhat a part of the misallocation in the intangible-intensive sectors is driven by the technologicalconstraints rather than by any sort of firm-level distortion. Also, similar to what we documentin the model, mean TFPR is also rising with intangible intensity in the data as well (see Figure9c). Finally, we document that the intangible-intensive sectors have higher sales-weightedprofit rate (see Figure 9e) and higher concentration as measured by HHI as well (see Figure 9dfor sales based HHI and Figure D.15 in Appendix D.3 for employment based HHI).41
6.2 Quantitative importance of IBTC: Steady State Comparison between 1980 &2007
In the previous section, we examined the implications of the heterogeneity in intangible in-tensity on firm dynamics and highlighted the underlying mechanism. Further, we providedempirical evidence to support these findings. Having assessed the ability of the model to pro-
41All results, except for the profit rates, are robust to the use of SIC3 level as sectoral classification, i.e., 101 sectorsin total (see Table D.17 in Appendix D.3). However, numbers of firms in a given sector decrease substantially atSIC3 level. Therefore, we focus on SIC2 classification for our main results.
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Figure 9: Intangible Intensity and Sectoral Differences:1980-1990
Slope = 0.00167***0
.01
.02
.03
Inta
ng
ible
In
ve
stm
en
t S
ha
re
0 5 10 15
Intangible Intensity
(a) Intangible Investment Over Sales
Slope = 0.00524***.32
.34
.36
.38
.4
sd
(tfp
r)
0 5 10 15
Intangible Intensity
(b) Dispersion in TFPR
Slope = 0.00561***8.28
8.3
8.32
8.34
8.36
8.38
me
an
(tfp
r)
0 5 10 15
Intangible Intensity
(c) Mean TFPR
Slope = 0.00198**.17
.18
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.2
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I
0 5 10 15
Intangible Intensity
(d) Herfindahl-Hirschman Index
Slope = 0.00088*.02
.03
.04
.05
Pro
fit
Ra
te
0 5 10 15
Intangible Intensity
(e) Weighted Profit Rate
Slope = 73.16**1000
1500
2000
2500
Sale
s
0 5 10 15
Intangible Intensity
(f) Average Sales
Note. Binscatter plot with intangible intensity (i.e., stock of intangibles divided by employment) on x-axis and ony-axis. Each blue dot represent mean of the x-axis and y-axis variables within equally-sized bin. The slopes of thelinear-fit provided in each plot is computed while controlling for year and sector fixed effects. The y-axes reportthe intangible intensity of the sector, defined by the ratio of intangibles to physical capital, in all figures at SIC2level. Plot a presents mean of the share of intangibles in sales. Plot b represents the standard deviation in TFPR.Plot c shows mean share of profits in sales (sales-weighted). Plot d shows HHI. All variable are averaged acrossthe sample period 1980-2010. Intangible share, physical capital share, profit-rate and TFPR are winsorized at 1and 99th percentile to remove outliers.
duce the results that are qualitatively consistent with the cross-sectional moments in the data,we move to the macroeconomic analysis of IBTC between the period 1980 and 2007. To es-timate the effect of a technological transformation along the same line as the one presented
33
before, we compare the model outcomes in two different steady states between the years 1980and 2007. As discussed earlier, the 1980 steady state is represented by our benchmark calibra-tion (i.e., average economy between the years 1980 and 1990) as discussed in Section 5, whereaswe compute the 2007 steady state using the estimated input shares between 2001-2007 (see Ta-ble 1 in Section 3.1.1), keeping the other parameters unchanged (i.e., their values remain sameas in the 1980 steady state). We consider the IBTC in the production function at firm-level asexogenous. A full parameterization for this exercise is described in Table D.11 in appendix.
In particular, we quantify the effect of the rise of intangible capital share in the US onthe dispersion in TFPR and concentration over the last three decades. These moments havereceived a particular attention in the literature recently, see, for instance, Grullon, Larkin, andMichaely (2019) and Autor, Dorn, Katz, Patterson, and Van Reenen (2017) for concentration,and Kehrig (2015), Caggese and Perez-Orive (2017), Hsieh and Klenow (2017), and Decker,Haltiwanger, Jarmin, and Miranda (2018) for rising dispersion in TFPR. Moreover, the modelalso offer an explanation regarding the rise in profit rates as documented by Barkai, 2016 andDe Loecker, Eeckhout, and Unger, 2020.
First, we plot the distribution of intangible intensity across firms in the model for differentsteady states. It is important to note that the model is able to capture the distributional shift inthe intangible intensity that is observed in the data (see Figure 10). Moreover, mean intangibleinvestments as a share of firms’ sales increase by a factor of more than two in the model and bya factor of three in the data. Second, we provide statistics for the percentage change in variousmeasures between the two steady states (1980 and 2007) in the model and compare that to thechanges observed in the data (see Table 5). We use two different sources of data: BDS andCompustat.
In particular, we observe an increase of 7.3% and 8.2% in average firm size in the modeland the data between 1980 and 2007, respectively. Similar increase in entrant size is observedas well. Next, we report the changes in the dispersion in TFPR, HHI and profit rate: First,the model predicts an increase of 9% in the the standard deviation of TFPR between 1980to 2007, which explains almost one-third of the variation that we observe in the data. Thisfollows from the nature of technological change that we estimated in section 3.1. The economyis moving from labor to intangible capital, meaning that the labor share went from 0.67 to0.595 whereas the intangible capital share went from 0.03 to 0.095. Therefore, there has beena shift from a relatively flexible input, labor, to an input, intangible capital, that is semi-fixedand highly distorted due to the presence of high adjustment costs. As discussed earlier, theadjustment cost does not allow the equalization of marginal products across firms and thus,of TFPR as well. Hence, when technology becomes more intangible-intensive, the dispersionin TFPR further increases. Taking this number at its face value tells us that the a substantialportion of the increase in dispersion of TFPR we observe in the data is due to the underlyingtechnological change that the US economy is undergoing.
Moreover, we observe that the model has predictions that are consistent with at least twoother broad trends that we are observing in the US economy. First, the model predicts anincrease of 27% in profit rates relative to the almost 37% increase that we find in the data. Thisis driven by a combination of two forces: better selection of firms as only highly productivefirms operates and the decline in the cost of hiring labor (i.e, wage). This is consistent with therecent literature which documents a rise in sales-weighted profit rates as mentioned before.
Second, our model predicts an increase in concentration by 26%, which is driven by the fact
34
Table 5: Results- Steady State Comparison
% ∆ 1980-2007
1980 S.S. 2007 S.S Model BDS Compustat ∆ Explained
Average firm size 20.5 22 +7.3 +8.2 +36 20%Average entrant size 5.6 8.1 +44 +11 - -ki/` 0.30 0.68 +126 - +310 -xi/y 0.03 0.08 +250 - +460 -Profit rate 0.18 0.23 +27 - +37 72%sd(TFPR) 0.24 0.26 +8.3 - +22 38%HHI 7.7e-4 9.7e-4 +26 - +39 66%
Notes. We report average size of firms and average size of incumbents with Business Dynamics statistics (BDS)data available online. For all the statistics reported for Compustat data, except for profit rate, we compute sectoraverages at SIC 2-digit and then weight it by sector sales share to compute yearly averages. For profit rate, wecompute sales-weighted average at SIC 2 digit level and then weight it by sector sales share to compute yearlyaverages.
that the economy selects highly productive firms into the production process. This is consistentwith the increase in concentration, measured by the HHI index, as shown by Autor, Dorn, Katz,Patterson, and Van Reenen (2017) and Grullon, Larkin, and Michaely (2019). It is importantto note that there is no change in the degree of competition (i.e., model operates under theperfect competition) in our framework but still our model can explain a substantial part of theincrease in concentration in the US economy. This in our view reinforce the widespread notionin industrial organization literature, that relying only on the HHI as a measure of competitioncan be misleading, as argued recently by Berry, Gaynor, and Scott Morton (2019).
Figure 10: Distribution of Intangible Intensity
-4 -2 0 2
Intanigble Intensity
0
0.1
0.2
0.3
0.4
0.5
De
nsity
1985
2005
(a) Model
-10 -5 0
Intanigble Intensity
0
0.05
0.1
0.15
De
nsity
(b) Data
Notes. Intangible intensity is defined as the stock of intangible capital over employemt. For distribution in thedata, the 1980 steady states represents the years between 1980 and 1990, whereas 2007 steady state represents theyears between 2001 and 2007.
To summarize our main results, we want to highlight that the quantities of the macro-trendsthat are explained by our mechanism may vary depending upon the measure of intangible cap-ital employed and its implied share with the chosen production function estimation technique.For instance, in our baseline results, we increase the input share of intangible capital by a fac-
35
tor of three (i.e., what we documented in section 3.1 with Olley-Pakes estimation). However,in our production functions estimations, especially the one performed with Ackerberg, Caves,and Frazer (2015) methodology, we documents that the intangible input share has increase byalmost a factor of two (from 0.07 to 0.14). The levels of the intangible input share are differ-ent across production function estimation techniques, but we can use insights from our mainresults to say something about the quantities of the macro-trends that can be explained if theintangible input share has doubled and not tripled since the 1980s (as in our baseline specifica-tion). In this scenario, we find that, by doubling the intangible input share from its 1980s level,the model can explain one-tenth of the rise in the firm size and dispersion in TFPR and almostone-third of the rise in the industry concentration and sales-weighted profit rates.
Therefore, we can say that the main result of the paper – that the substantial portions ofthe macroeconomic changes that we witness in the economy can be explained by the changingnature of the production technology – remain there, regardless of the production function es-timates employed, although the exact quantities of the macro-trends that can be rationalizedby our mechanism may vary with both the levels of input shares and the estimated changes inthe production function over the past three decades.
7 Discussion and Robustness
In the last section, we have discussed the implication of the rise of intangible input share in theproduction function on the firm and investment dynamics in the context of the model. In thissection, we discuss the implications of the modeling choices and provide further robustnesschecks on the production function estimation methodology.
7.1 Model
The model is designed to capture some salient features of firms’ investment behavior in the US.We focus on two main points: first, the model entails a relatively flexible production structure,in terms of returns to scale and input shares, which allows us to measure precisely the produc-tion technology in the data. This means that, despite the fact that we assume a Cobb-Douglasfunctional form, we can leave all the other parameters free. This is particularly relevant forour analysis because so far the importance of this new capital in production has not been rig-orously investigated. Second, since we allow for a general capital adjustment cost function,this leaves the model sufficiently flexible to be able to generate an investment distribution inthe model that replicates the one in the data. We see this as first step to pin down rigorouslythe properties of this relatively unfamiliar input of production.
Moreover, we want to emphasize that we interpret the model as a stationary version of onewith growing productivity. This means that in our world investing in intangible is a profitablebusiness for US firms despite the potentially higher adjustment costs. Therefore, the way wethink about this environment is as if we are in an economy, where every period new andbetter technological vintages are available but, from time to time, these new vintages entailhigher adjustment costs, e.g., intangible-intensive technologies. This underlying conceptualframework implies that in a de-trended world it could appear as if the adopted technology isless valuable from the point of view of the firm that adopts it; however, this is just a byproduct
36
of a convenient transformation of the economy and not a sensible economic result.42
7.2 Alternative Structural Changes in the US economy
In this section, we discuss alternative mechanisms that have been put forward by the literatureand evaluate their macroeconomic implications with a particular emphasis on the resourcemisallocation, firm size and industry concentration. Finally, we compare these results to ourbaseline results that are mentioned in Section 6.
The Rise of Fixed costs. Recently, a number of papers have argued that the rise of fixedcosts in production is responsible for the underlying changes the US economy has witnessedin the past three decades (e.g., see De Loecker, Eeckhout, and Mongey, 2019, Korinek and Ng,2018, De Ridder, 2019, Hsieh and Rossi-Hansberg, 2019, and Sandström, 2020). We comparethe effects of the rise in fixed cost to that of IBTC.
In order to highlight the macroeconomic effects of the rising fixed costs in our framework,we increase c f , while keeping everything else constant. The value of parameters are providedin Table D.18. The values of the parameters in the benchmark economy (i.e., steady state in1980s) is kept same as the one used in the baseline results. In order to calibrate the value ofc f in the 2007 steady state, we match the change in the ratio of fixed cost to output that weobserve in the data between 1980 and 2007. In Compustat, fixed costs (overhead costs) aremeasured by XSGA (see section A for more details). The ratio of total fixed costs to totaloutput has increased by 45% approximately. We estimate c f = 3.26 in 2007 steady state, thatgives us a rise of approximately 45% in total fixed costs to total output in the model. We showthat, contrary to our baseline results, the new steady state shows a counterfactual decline infirms’ average intangible capital invest as a share of their output. Furthermore, also the levelof (mis)allocation of resources in the economy remains unchanged in the new steady state.Taking stocks, the rise in fixed costs, while explaining some of the trends in the US economy,fails to explain the main objects of interest of this paper such as the rise in intangible capitaland the rise in (mis)allocation.
The Decline in the Relative Price of Intangible Investments. In the baseline model, weassumes that the relative price of intangible investments, pi = 1, and it is constant over time.However, as emphasized by Zhang (2019) , we know that the relative price of intangible capitalhas declined steadily over the years. The results are shown in Table D.21. In our model, thedecline in the relative price of intangible capital fails to explain all the trends we observe in theUS economy. In this scenario firms become smaller and invest less. The average profit in theeconomy declines due to a relaxation in the selection process. Moreover, both concentrationand (mis)allocation decline. We conclude that this channel, relative to IBTC, has counterfactualimplications for the main objects of interest of this paper.
The Decline in the Interest Rate. In the baseline model, we assume that the real interestrate, R, is constant across time. However, it has declined in the past three decades. This channelhas been put forward by Liu, Mian, and Sufi (2019). In our model, the risk-free interest rateis directly calibrated. Therefore, in order to compare the implications of such a decline withthe implications given by IBTC we just recalibrate the risk-free interest rate from 5% to 2%keeping all the other parameter fixed. This change is consistent with measurement by Rachel
42The quantitative study of a non-stationary economy, growing around an endogenous balanced growth pathwith technology acquisition/diffusion, is a significantly more difficult task that, despite being of its own interest,is outside the scope of this project.
37
and Smith (2015). Table D.22 shows our calibration. The results are shown in Table D.21. In ourmodel, the decline in the risk-free interest rate fails to explain all the trends we observe in theUS economy. In this scenario firms become smaller and invest less. The average profit in theeconomy declines due to a relaxation in the selection process. Moreover, both concentrationand (mis)allocation decline. We conclude that this channel, relative to IBTC, has counterfactualimplications for the main objects of interest of this paper.
7.3 The Impact of IBTC on Market Power and Policy Implications
In our framework, as production technology becomes more intangible-intensive, firms investmore in an input that entails higher adjustment costs. Despite the fact that this technologicalchange raises market concentration, firm size and sales-weighted profit rates, resources arestill allocated efficiently across firms. The observed rise in misallocation in the model is dueto technological constraints, and therefore, the decentralized equilibrium allocation is coin-cides with the one provided by social planner. Our paper suggests that a substantial part ofthe macroeconomic changes that we witness in the US economy could be efficient (similar toLashkari, Bauer, and Boussard, 2018 and Hsieh and Rossi-Hansberg, 2019), therefore, we differfrom a number of recent contributions that instead focus on distortionary consequences of thetechnological change (e.g., Korinek and Ng, 2018, De Loecker, Eeckhout, and Unger, 2020, andSandström, 2020, Baqaee and Farhi, 2020, Aghion, Bergeaud, Boppart, Klenow, and Li, 2019,and De Ridder, 2019).
However, these two different accounts of the recent trends are not mutually exclusive. Forinstance, consider a slightly different version of our baseline model. Instead of assuming thatfirms produce the same good, we can allow firms to produce different varieties and then in-troduce a final good producer that aggregate all the varieties in the economy. In order tohave variable markups, we can use Kimball aggregator that allows for a monotonically in-creasing relationship between markups and relative firm size (see Edmond, Midrigan, andXu, 2018). Now, we can rely on the results of Section 6 to draw insights about the macroe-conomic response of this modified model to the rise of intangible capital. In this new frame-work, markups are positively correlated with firm size. Therefore, a technological change thatfavors larger firms would shift market shares toward high-markup firms and away from low-markup firms. These patterns are in line with those documented by Baqaee and Farhi (2020)and De Loecker, Eeckhout, and Unger (2020). Moreover, in this framework, the labor sharewould decline even more as the average markup increases. The measured misallocation (dis-persion in TFPR) would be magnified by the rise in the dispersion in markups on top of themisallocation generated by the IBTC.
As far as policy implications are concerned in our modified framework with markups, theimplementation of the first-best allocation would coincide with the allocation in our baselineframework. As a consequence, while we see clear margins of policy intervention in the realworld, our framework cautions the policymakers to acknowledge that a sizable fraction of themacroeconomic trends that we observe in the US economy could be a by-product of a changingproduction technology.
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8 Conclusion and way forward
In the last four decades, firm-level investment in intangible capital has dramatically increasedin the US. However, we still know little about its properties and its cross-sectional and as wellas aggregate implications are not well understood. In this paper, we made a step forward toanswer these questions.
First, we have shown that intangible capital is an important input of production and itsinput share has witnessed a threefold increase since the 80s. We interpret these findings as aparadigm shift in the production technology of US firms, e.g., see the importance that softwareand other intellectual properties products have increasingly gained in our economies. we re-fer this as the intangible investment biased technological change. Finally, consistent with theprevious literature, our estimations find a decline in the labor input share at firm-level. Thissuggest a potential substitution of labor with intangible capital at firm-level.
Second, we have highlighted some of the properties of intangible capital, particularly, thatit entails much higher adjustment costs compared to physical capital – particularly fixed ad-justment costs. These results are consistent with the view that such investment are plagued byinherent indivisibilities (for example, investments in production management software requirelarge investment beforehand) and are more sunk. Moreover, in line with the what we find inthe data, the presence of high adjustment costs make the average product of intangible capitalin the model more dispersed relative to the one of physical capital. This implies that intangiblecapital tends to be more misallocated than physical capital due to technological constraints.
We draw cross-sectoral predictions, from the structural model, implied by different sec-toral intangible intensities. The model predicts that the dispersion in TFPR, average account-ing profit over sales, average productivity and concentration increase in intangible intensity,whereas, physical investment over sales decreases with intangible intensity. We find supportfor these predictions in the data.
Finally, we documented the quantitative implications of the intangible biased structuraltransformation for the overall US economy. We found that the doubling of the intangible shareof production, as seen in the data, can explain a sizable part of the observed rise in TFPRdispersion and in accounting profits. Moreover, it is broadly consistent with rising industryconcentration. Therefore, we interpret our findings as an alternative explanation to some of thetrends documented in the US economy that is complementary to the literature that emphasizethe rise of market power. The study of the interaction of these two forces is an exciting avenuefor future research.
In the future, we want to understand the macroeconomic implications of the rise of intan-gibles in the short-run. It is important to understand the firm behavior, investment dynamics,and its implication on firm size, dispersion in TFPR, concentration and labor share in the im-mediate aftermath of the shock. To do so, in the future, we plan to simulate the transitionaldynamics between the two steady state.
Given the findings of this paper, a natural next step is to understand the implication ofintangible investment biased technological change on the industry structure. Moreover, ex-amining the origins of such shift in the inputs of production is a potential avenue of futureresearch. Finally, We think that further investigation into the substitution between labor inputand intangibles is a potential avenue for future research.
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42
A Appendix: Data: Main Sample, Variables, and Summary Statis-tics
In this section of the appendix we explain how the main sample has been cleaned, we showthe main variables and the summary statistics.
A.1 Main Sample Cleaning
We use Compustat from 1965 to 2015. We drop all the firms whose Foreign Incorporation Code(FIC) is not equal to USA. Then, we use linear interpolation on SALE, COGS, XSGA, EMP, AT,PPEGT, PPENT, INTAN, XRD.43 For data quality, we interpret as mistakes if SALE, COGS,EMP, AT, PPEGT or PPENT are negative, zero or missing and we drop these observations. IfXSGA, INTAN or XRD are negative we drop them, if they are missing we assume they arezero. We exclude utilities (SIC codes between 4900− 4999) because they are heavily regulatedon prices. We also exclude financials (SIC codes between 6000− 6999) because their balancesheets are dramatically different from other firms. To get a real measure of SALE, COGS,XSGA we deflate them with the GDP deflator, we deflate investment in physical capital by theinvestment good deflator and we deflate investment in intangible capital by the IPP deflator.44
The table below presents a few basic summary statistics for a few leading variables used in ouranalysis.
Our baseline production function estimation relies on value added as the measure of firmoutput. We follow Bloom and Van Reenen (2007) and Imrohoroglu and Tüzel (2014) to com-pute value added in Compustat. In particular, we subtract costs of materials from firms’ sales.However, Compustat does not separately report cost of material for all firms. It only providesCOGS (i.e., sum of material and labor costs), and for a subset of firms labor cost is reportedas well. We compute firm-level wages using labor costs and total number employees. We re-place missing value with the sector-level averages. Using these imputed labor costs, we cancompute material costs and, therefore, value added for all firms in the sample.
Table A.1: Summary Statistics (1980-2007)
Variable No. Obs. Mean p10 p25 p50 p75 p90
Sales SALE, y 204,461 1,627 6.033 30.50 157.0 715.3 2,861Cost of Goods Sold COGS, v 204,461 1,131 3.920 18.43 97.57 471.2 1,930Employment EMP, ` 204,461 7.054 0.0410 0.161 0.816 3.736 14Physical capital kp 204,461 608.1 0.997 4.248 24.12 152.9 824.2Intangible capital ki 204,461 321.1 0 0.0503 4.128 43.97 308.3
Note. Summary statistics of cleaned Compustat dataset between 1980 and 2007. For each variable we list the Compustat acronymand the associated notation (in levels) used throughout the manuscript. All variables but employment are in millions of US Dol-lars deflated with the base year 2012. Employment is measured by the number of workers within each firm and is in thousands.
43This practice affects a small portion of observations and it does not affect our results.44Deflators are taken from the NIPA tables.
43
A.2 Main Variable: Intangible Capital
In this section, we document how to construct in the Compustat our main measure and threeother measures (for robustness) of firm-level intangible capital. We deflate all components ofintangible capital with IPP deflators provided by Bureau of Economic Analysis (BEA) (ImplicitPrice deflators table 1.1.9.).
For externally bought intangible capital, Kexti,t =
Intani,tIPP De f lator . The knowledge capital is con-
structed krdi, f t = (1− δs)krd
i,t−1 +R&Di,t
IPP De f lator , where depreciation rate δs is sector-dependent taken
from Ewens, Peters, and Wang (2019). The organizational capital korgi, f t = (1 − 0.20)korg
i,t−1 +
γsSGAi,t
IPP De f lator , where γs is the sector-dependent share of intangible investment taken fromEwens, Peters, and Wang (2019).
Therefore, our four measures of intangible capital are going to be as follows,
1. Baseline: krdi,t + Kext
i,t ;
2. Baseline - Goodwill: krd + Kexti,t − Goodwillit;
3. R&D + SGA: krdi,t + Korg
i,t ;45
4. R&D + SGA + Externally Purchased: krdi,t + Kext
i,t + Korgi,t . 46
In order to get our main main measure of aggregate intangible capital, we closely followKoh, Santaeulalia-Llopis, and Zheng (2016). Data on nominal gross value added are takenfrom the National Income and Productivity Accounts (NIPA) Table 1.14 (line 17). Data oncompensation of employees are taken from the NIPA Table 1.14 (line 20). The main data sourcefor investments are Tables 3.7(I,S, and E).
Further, it is well established that BEA’s measure of intangible capital is an incompletemeasure (Corrado, Hulten, and Sichel, 2009 and McGrattan and Prescott, 2010). We rely onCorrado, Hulten, and Sichel, 2009 and update for intangible items that are not included innational accounts. These series of intangible investment consist of (1) Finance and insurancenew product development, (2) Design, (3) Brand, (4) Training, and (5) Organizational capital.
The aim of this section is to compare our firm-level measures and the aggregate measuresand validate them. We show in Figure A.1 that the intangible investment share of sales isrising over time for all the four measures from Compustat. This rise is consistent with what isdocumented in the aggregate data provided by the BEA and by Corrado, Hulten, and Sichel(2009). Between 1980 and 2015, the intangible investment share of sales among the Compustatfirms increased by a factor of ten in our baseline measure and by a factor of five in the secondmeasure (i.e., baseline - goodwill). The third measure increased by a factor of 2 and the finalmeasure increased by a factor of five.
The steep rise in intangible investment is potentially driven by the fact that, first, the Com-pustat captures the largest firms in the economy and we know that these are the firms thattend to be more intangible intensive. Secondly, Compustat sectoral composition is differentform that of the aggregate economy.
In Figure A.2, we use the BEA sectoral shares to aggregate the intangible investment sharein the Compustat economy. It is clear that our intangible capital measure is in line with the
45This measure only uses information from income statement.46Measure as in Peters and Taylor (2017).
44
aggregate data and does not over predict the intangible investment at the firm-level. A moredetailed discussion on these issues, with information at more disaggregated level, is providedin the Online Appendix.
Figure A.1: Intangible Investment Share in Output: 1980-2015
1980 1985 1990 1995 2000 2005 2010 2015
0
0.05
0.1
0.15
0.2
Baseline
Baseline - Goodwill
R&D + SG&A
R&D + SG&A + Intan (PT)
(a) Intangible Investment Share (Compustat)
1980 1985 1990 1995 2000 2005 2010 2015
0
0.05
0.1
0.15
0.2
BEA
BEA + CHS (2009)
(b) Intangible Investment Share (BEA & CHS)
Note. The left figure shows the rise of total intangible investment to total sales in the Compustat dataset for all of the fourmeasures. The figure on the right shows the total intangible investment to GDP from the National Accounts. The time-series areplotted from 1980-2015 and they all show a sizable increase
Figure A.2: Intangible Investment Share in Output: 1980-2015
1995 2000 2005 2010 2015
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18 Baseline
Baseline - Goodwill
R&D + SG&A
R&D, SG&A, and Intan (PT)
BEA
BEA+ CHS (2009)
Note. The left figure shows the rise of total intangible investment to total sales in the Compustat dataset for all of the fourmeasures. The figure on the right shows the total intangible investment to GDP from the National Accounts. The time-series areplotted from 1980-2015 and they all show a sizable increase
Next, we look at which component of intangible capital, i.e., externally acquired intangiblecapital or internally produced intangible capital, is the main driver of this increase in the in-
45
tangible capital. Figure A.3 presents the evolution of the two components of intangible capitalas a share of total intangible capital over time.
Figure A.3: Decomposition of Intangible Capital into different Components
.2
.4
.6
.8
1
INTA
N/In
tang
ible
Cap
ital
1980 1990 2000 2010Year
AllConsumerManufacturingHi-TechHealthare
(a) Baseline Measure
0
.2
.4
.6
.8
R&D
/Inta
ngib
le C
apita
l
1980 1990 2000 2010Year
(b) Alternative Measure
Note. Figure (a) shows the evolution of externally acquired intangible capital as a share of total intangible capital(INTAN f t/(INTAN f t + krd
i, f t)) across the SIC sector classification over time. Figure (b) shows the evolution of internally pro-
duced intangible capital as a share of total intangible capital (krdi, f t/(INTAN f t + krd
i, f t)) across the SIC sector classification overtime.
It can be seen that the share of externally acquired intangible capital in the total intangiblecapital has increased over the period. Whereas, internally produced intangible capital hasseen a steady decline over the period. Therefore, we can conclude that the rise of intangiblecapital in the period of interest is entirely due to the rise of externally acquired intangiblecapital. The decline of R&D investment as the share of total intangible investment is in linewith the aggregate data provided by BEA and the findings of Koh, Santaeulalia-Llopis, andZheng (2016). Furthermore, Corrado, Hulten, and Sichel (2009) find that the major rise inthe intangible investments in the US economy is driven by the investment in software andorganizational capital.
A.3 Other variables
Figure A.4 shows the joint evolution of the intangible capital share in production, of concen-tration (HHI), and of (mis)allocation, i.e., the standard deviation of TFPR.
46
Figure A.4: Intangibles, (Mis)allocation & Concentration: U.S. Compustat
Concentration
Misallocation
Intangible InvestmentShare
1
6
11
Inve
stm
ent/S
ales
1
1.2
1.4
1.6
1.8
2
2.2
HH
I, sd
( TFP
R)
1980 1985 1990 1995 2000 2005Year
Note. The figure presents the evolution of the input share of intangible capital estimated with Olley-Pakes (black dashed-line),HHI index (orange line) and dispersion in productivity (blue line) over time. For all variables initial level (1980) is normalized toone.
It can be seen that all three measure show an important increase since the 1980, as alreadypointed by a vast literature.
Moreover, we present our measures of economy profits, with and without the inclusion ofintangible capital in the computation. Results are presented in figure A.5.
Figure A.5: Weighted Profit-Rates: U.S. Compustat
.04
.06
.08
.1
.12
Prof
it-Ra
te
1980 1985 1990 1995 2000 2005Year
.04
.06
.08
.1
.12
Prof
it-Ra
te
1980 1985 1990 1995 2000 2005Year
Note. Figure (a) presents the evolution of our measure of profit-rate, see equation 4, weighted by sales over time. Figure (b)shows the profit rate measure that is consistent with the one that is generally used in the literature. Orange line is the linear fit.
Both measures are increasing over the sample period. It can be seen that, once accountingfor the rise of intangible capital, the profit-rate rate shows a lower increase over time, however,not enough to account for the overall increase as observed in the US economy.
Finally, we show the evolution of the average firm size in the US economy from both theCompustat and the BDS data. From Compustat we show two measure of size, sales and em-ployment, whereas, from BDS only employment is available. Both sources show a robust in-crease in firm-level size.
47
Figure A.6: Average Firm Size: U.S. Compustat
1
1.2
1.4
1.6
Empl
oym
ent
1980 1985 1990 1995 2000 2005Year
(a) 1980-1990
.8
1
1.2
1.4
1.6
1.8
Sale
s
1980 1985 1990 1995 2000 2005Year
(b) 2000-2007
Note. The data source is Compustat. Figure (a) presents the evolution evolution of average employment. Figure (b) shows theevolution of average sale. Orange line is the linear fit.
Figure A.7: Firm Size BDS
20
21
22
23
24
Empl
oym
ent
1980 1990 2000 2010Year
Note. The data is provided by Business Dynamics Statistics. The figure shows the evolution of average firm-level employment.
48
B Appendix: Estimating Output Elasticities
In this section of the appendix, we present the different estimation techniques adopted to findthe production function elasticities in the data. Finally, we explain how we constructed vari-ables relate to the production function estimation.
B.1 Production Function Estimation
To address the endogeneity issue in the production function estimation, we use three main es-timation techniques that has been proposed in the empirical industrial organization literature:cost shares approach, Olley and Pakes (1992) structural estimator and dynamic panel data ap-proach. Finally, we provide estimates from Ackerberg, Caves, and Frazer (2015) structuralestimator. We discuss their positive and negative aspects, and their validity under slightly dif-ferent assumptions.
Cost Shares: This approach has been prominently adopted in Foster, Haltiwanger, andSyverson (2008) and it relies on two main assumptions: the presence of constant return to scaleand that all inputs are variable. This is the reason why we compute, as in Foster, Haltiwanger,and Syverson (2008) for example, the average cost share across the sub-sample periods to try toeliminate the cross-sectional deviations due to adjustment costs. Finally, an extra requirementto apply this method in the presence of three inputs is the possibility to calculate the return onphysical capital. To do so we follow Barkai (2016), and we use the following equation:
rt = (it −Etπt+1) + δp, (19)
where it is the nominal interest rate, Etπt+1 is expected inflation at time t and δp is the de-preciation rate of physical capital. The nominal interest rate is the annual Moody’s SeasonedAaa Corporate Bond Yield, whereas, inflation is calculated from the annual growth rate of theInvestment Nonresidencial Price Deflator and depreciation rate is calibrated to δp = 0.07, as inthe rest of the paper.4748
Dynamic Panel Data: The estimation of the production function with the dynamic paneldata approach follows the spirit of Cooper and Haltiwanger (2006) The production function is:
y f t = eait kαp, f tk
ωi, f t`
νf t, (20)
where y f t is real output for firm f at time t, ait is productivity, ki, f t is the stock of intangiblecapital, kp, f t is the stock of physical capital and ` f t is employment. The productivity ait iscomposed by At + z f t, where At is the time varying aggregate shock and z f t is the idiosyncraticshock whose low of motion is given by:
z f t = ρzz f t−1 + η f t, η f t ∼ N(0, σz), (21)
47https://fred.stlouisfed.org/series/AAA48https://fred.stlouisfed.org/series/A008RD3Q086SBEA. We estimate an AR(1) on the annual growth rate of
the Investment Nonresidential Price deflator process and define the contemporaneous expected inflation Etπt+1 =µ + ρπt.
49
where ρz is persistence of productivity process and η f t is the innovation that is unobservableto the firm. Firms adjust both their capital stocks and their employment according to theirproductivity and this create a endogeneity problem that bias the estimates of a simple OLS.Therefore, we quasi-differentiate the logs of equation 20 which yields:
log(y f t) = ρz log(y f t−1) + α(kp, f t − ρzkp, f t−1) + ω(ki, f t − ρzki, f t−1)
+ ν(` f t − ρz` f t−1) + At − ρa At−1 + η f t.(22)
We estimate equation 22 with a generalized method of moments (GMM), where lagged in-tangible capital, lagged physical capital, lagged employment and lagged output are used asinstruments as they are uncorrelated with ηit.
Olley and Pakes Estimator: To implement the Olley and Pakes method in Compustat, wefollow the lead of Bloom and Van Reenen (2007), Imrohoroglu and Tüzel (2014) and De Loecker,Eeckhout, and Unger (2020). Particularly, we assume that employment is a fully flexible, andwe use investment in physical capital as control variable.49 The production function is:
log(y f t) = α log(kp, f t) + ω log(ki, f t) + ν log(` f t) + zit + ε f t. (23)
The solution to the firm’s dynamic optimization problem (that is consistent with our theo-retical framework) results in a equation for physical investment xp,it, given by:
log(xp,it) = φt(log(kp, f t), log(ki, f t), z f t, d f t), (24)
where function φt is strictly increasing in productivity z. The variable d f t captures output andinput market conditions that generate variation in factor demand across firms, conditional onthe level of productivity and both capital stocks. In our baseline case of perfect competitionin both input and output markets, d f t adds no extra information. However, as we discussbelow, a slight departure from this assumption, d f t becomes crucial to estimate the outputelasticities. To allow for a non-parametric inversion of the investment function, we need strictmonotonicity assumption – investment is monotonically increasing in productivity. Moreover,we require productivity to be the only unobservable entering the function of investment andfinally, we need that kit is determined in period t− 1. The inversion of the investment functionproduces the following control function:
z f t = ht(log(kp, f t), log(ki, f t), log(xp, f t), d f t). (25)
Given 25, we define an auxiliary function given by:
Φ f t = α log(kp, f t) + ω log(ki, f t) + ν log(` f t) + h(·). (26)
Using equation 23 and 26, we get the following first stage regression:
log(y f t) = ν log(` f t) + Φ f t + ε f t, (27)
49Results are robust to the use of investment in intangible capital as control variable. However we do not use itas main specification since, as explain in the main body of the text, this investment is more often zero and therefore,we are left with a smaller sample for the estimation. This increases the error in the estimation.
50
where we approximate Φ f t with a nth order polynomial series in physical capital, intangiblecapital, and investment in physical capital. The partially linear equation 27 is estimated withOLS. This gives us a consistent estimate of ν. However, this does not produce unbiased es-timates for α and ω. Achieving this requires a second step to estimate survival probabilities,which will then allow us to control for selection bias in the productivity distribution. Remem-ber, firm will choose to stay in the market if its productivity is greater than some threshold, z f t,that depends on ki, f t and kp, f t. The probability of survival in period t depends on z f t−1 andz f t−1, both capitals stocks, and investment at time t?1. In our implementation, we estimatethe probability of survival by fitting a probit model of χ f t (the exit rule) on i f t−1, ki, f t−1, andkp, f t−1, as well as on their squares and cross products. The predicted probability in the modelis Psurvivalt .
log(y f t)− ν log(` f t) = α log(kp, f t) + ω log(ki, f t)+
gt(Φ f t−1 − α log(kp, f t)−ω log(ki, f t), Psurvivalt) + ξ f t + ε f t.(28)
In the third step, we use nonlinear least squares to fit Equation 28, where g(.) is approxi-mated with a second-order polynomial in Φ f t−1 − α log(kp, f t)− ω log(ki, f t) and Psurvivalt . Thefunction g(·) is similar to the inverse of Mills’ ratio that is included in two-step sample selectionmodels. This step gives us consistent estimates of α and ω.
B.2 Additional Results from the Main Specification
We provide the estimates for physical capital. Further, we provide error bands for the OProlling window estimation.
Figure B.8: Trends in Physical capital Input shares
0
1
2
3
4
Phys
ical
Inpu
t sha
re
1980 1985 1990 1995 2000 2005Year
Note. We present physical capital elasticity estimated with OLS (orange line - triangles), Cost shares (black line - dashed), Olley-Pakes (green line - diamonds) and Dynamic panel data (blue line - solid). We use a ten-year rolling windows to compute theinput shares over time.
51
Figure B.9: Trends in Input shares: U.S. Compustat
-.05
0
.05
.1
.15
.2
Phys
ical
Inpu
t sha
re
1980 1985 1990 1995 2000 2005Year
CI-95%Point-Estimate
(a) Intangible Capital Input Share
.55
.6
.65
.7
Phys
ical
Inpu
t sha
re
1980 1985 1990 1995 2000 2005Year
(b) Labor Input Share
.2
.25
.3
.35
.4
Phys
ical
Inpu
t sha
re
1980 1985 1990 1995 2000 2005Year
(c) Physical Capital Input Share
Note. We present output elasticities estimated with OLS (orange line - triangles), Cost shares (black line - dashed), Olley-Pakes(green line - diamonds) and Dynamic panel data (blue line - solid). We use a ten-year rolling windows to compute the inputshares over time.
B.3 Robustness of Production Function Estimation
Here we present a different estimation technique and some alternative specifications to the onepresented in the main text. Then, we discuss the validity of our main results.
PFI - Ackerberg, Caves, and Frazer (ACF): The estimation of the production function withthe ACF approach follows the method mentioned in Ackerberg, Caves, and Frazer (2015).Contrary to the Olley-Pakes method, where coefficients of free inputs are estimated in thefirst stage of estimation, here, instead, the input coefficients are all estimated in the secondstage. However, the first stage will still be important to net out the error in the productionfunction. We use investment as a proxy, similar to Olley-Pakes method, to control for firm-levelunobserved productivity. The ACF procedure is robust to labor adjustment costs. Moreover,this method does not require to adjust for sample selection and instead relies on a unbalancedpanel.
The first stage OLS regression is
log(y f t) = Φ f t(log(` f t), log(ki, f t), log(kp, f t), log(xp, f t), d f t) + ε f t, (29)
52
where we approximate Φ f t with a second-order polynomial series in physical capital, intan-gible capital, labor and investment in physical capital. This first stage estimation results in anestimate for Φ f t. we use the estimated Φ f t to infer productivity z f t(α; ω) and innovations inproductivity ξ f t(α; ω). Finally, in the second stage, we can use the following moment condi-tions to estimate the two capital elasticities:
E(ξ f t(α; ω)× [ki, f t, kp, f t, ` f t−1]′) = 0. (30)
The results are presented in Table B.2. The production function estimated with the ACFestimator presents very similar patters to the one presented in the main specification. Partic-ularly, it can be observed that also this estimator produces a dramatic increase in intangiblecapital share, which still happens mostly at the expense of the labor input share.
Table B.2: Production Function Estimates with ACF method
log(y f t) log(y f t)
1980-1990 2000-2007
log(kp, f t) 0.270 0.299(0.044) (0.062)
log(ki, f t) 0.073 0.115(0.011) (0.015)
log(` f t) 0.66 0.576(0.014) (0.154)
Obs. 11523 8354
Notes: Production function estimates with ACF. log(kp, f t) is physical capital, log(ki, f t) is intangible capital, log(l f t)is the employment, and log(y f t) is value added, i.e., sales minus material costs. Material costs are not reported inCompustat since they are bundled together with labor costs in the expenditure voice Cost of Goods Sold. Therefore,to extract material costs we follow Bloom and Van Reenen (2007). Standard errors are in parentheses. We controlfor year fixed effects for each specification.
Input and Output Price Variation: As discussed in De Loecker, Goldberg, Khandelwal,and Pavcnik (2016), and De Loecker, Eeckhout, and Unger (2020), usually in firm-level datasets,we observe revenues and expenditure instead of quantities of output and input used. How-ever, the Cobb-Douglas production function is written in terms of quantities, therefore, theestimation procedures that do not use data on quantities and instead relies on revenue andexpenditure data are prone to multiple problems. For instance, in the presence of markup het-erogeneity, the error term would include output price variations and thus the estimated outputelasticities are biased. In particular, in our case, the structural error term in the production es-timation equation contains output and input prices,
z f t + Pf t − αPp,t −ωPi, f t, (31)
where Pf t is the firm specific output price, Pp,t is user costs of physical capital that is sameacross firms within a industry and Pi, f t is the user cost of intangible capital that varies across
53
firms. This specification allows for imperfect competition in the market of intangible capital.In the case of our baseline specification, the inclusion of the variable d f t should capture therelevant output and input market forces that generate differences in output and input price.As mentioned in De Loecker, Eeckhout, and Unger (2020), controlling for proxies for firmmarket power such as market shares can give us unbiased estimates of output elasticities.
We consider market share, measured at various level of aggregation (2 and 3 digit SICsector), to take into account additional variation in output and input markets (market of in-tangible capital). We refer to De Loecker, Goldberg, Khandelwal, and Pavcnik (2016) for moredetailed discussion on these controls and their efficacy under different model assumptions.
However, we do not rely on one estimation to guide our results on the input share of in-tangible capital. We do multiple robustness check to show that our inability to control forproducer level input and output prices do not change the baseline results alot.
The results are presented in Table B.2. The production function estimated with the ACFestimator presents very similar patters to the one presented in the main specification. Partic-ularly, it can be observed that also this estimator produces a dramatic increase in intangiblecapital share, which still happens mostly at the expense of the labor input share.
Table B.3: Production Function Estimates: ACF with Output and Input market controls
log(y f t) log(y f t)
1980-1990 2000-2007
log(kp, f t) 0.295 0.303(0.02) (0.001)
log(ki, f t) 0.076 0.141(0.001) (0.001)
log(` f t) 0.644 0.510(0.03) (0.003)
Obs. 11523 8354
Notes: Production function estimates with ACF. log(kp, f t) is physical capital, log(ki, f t) is intangible capital, log(l f t)is the employment, and log(y f t) is value added, i.e., sales minus material costs. Material costs are not reported inCompustat since they are bundled together with labor costs in the expenditure voice Cost of Goods Sold. Therefore,to extract material costs we follow Bloom and Van Reenen (2007). Standard errors are in parentheses. We controlfor year fixed effects for each specification.
Limitations of Investment as Proxy: In Olley-Pakes estimation, it is assumed that thereare no other firm-specific unobservables (other than the productivity shock z f t) that affectinvestment demand. As mentioned in Ackerberg, Caves, and Frazer (2015), for instance, itrules out unobserved capital adjustment costs that vary across firms, as well as unobserved,firm-specific shocks to investment prices. Moreover, Olley-Pakes does not allows serially cor-related, unobserved heterogeneity across firms in prices of labor or intermediate inputs. Thereason being that the unobserved input price (wage or material price) shocks that are seriallycorrelated over time, are linked to the marginal revenue product of capital at t + 1, and thusfirms’ investment at time period t is correlated to price shocks.
54
Table B.4: Production Function Estimates: ACF with Material Expenditure as Proxy
log(y f t) log(y f t)
1980-1990 2000-2007
log(kp, f t) 0.306 0.292(0.013) (0.002)
log(ki, f t) 0.0740 0.134(0.003) (0.000)
log(` f t) 0.620 0.523(0.030) (0.002)
Obs. 11523 8354
Notes: Production function estimates with ACF. log(kp, f t) is physical capital, log(ki, f t) is intangible capital, log(l f t)is the employment, and log(y f t) is value added, i.e., sales minus material costs. Material costs are not reported inCompustat since they are bundled together with labor costs in the expenditure voice Cost of Goods Sold. Therefore,to extract material costs we follow Bloom and Van Reenen (2007). Standard errors are in parentheses. We controlfor year fixed effects for each specification.
Ackerberg, Caves, and Frazer (2015) gave a solution to this problem. First, they considera value added production function in the sense that the intermediate input m f t does not enterthe production function to be estimated. There interpretation of this is that the gross outputproduction function is Leontief in the intermediate input, where this intermediate input isproportional to output. The approach relaxes assumptions typically made in applications ofOlley-Pakes. The model allows for exogenous, serially correlated, unobserved firm-specificshocks to the price of labor, or firm-specific unobserved adjustment costs to the labor input.Finally, as discussed above, it also allows the labor input to have dynamic effects. The mainassumption is that the material costs is monotonic in firm productivity and apart from pro-ductivity, labor demand and both predetermined capital stocks, there are no other sources ofunobserved heterogeneity in material demand. We follow De Loecker, Eeckhout, and Unger(2020) and include firm’s sales share to allow for input and output market variation. The re-sults are presented in Table B.4 and it is clear that the results are robust and intangible inputshare has risen significantly in the last three decades.
Measurement Error in Intangible capital: In principle, the measurement of the stock ofintangible capital may contain measurement error. This is because of many simplification as-sumption that we make to have its measure in the first place. Therefore, it is important tounderstand the nature of the measurement error and potential biases it may generate in theproduction function estimation. The error may arise due to variation in the investment pricedeflators, depreciation rates across firms. Therefore, aggregation over different vintages andheterogeneous asset types within the firm generate measurement error. Also, perpetual in-ventory method, that we use to build our stock of intangible capital requires, initial stock ofcapital and could be source of potential measurement error. In this case, our measure of intan-gible capital can be written as
log(ki, f t) = log(k∗i, f t) + ζkif t, (32)
55
where log(k∗i, f t) is the true stock and ζkif t is the classical measurement error in capital stock, such
that E[ζkif t] = 0: However, it can be serially correlated over time. In this case, the production
function would take the following form
log(y f t) = α log(kp, f t) + ω(log(k∗i, f t) + ζkif t) + ν log(` f t) + z f t −ωζki
f t + ε f t, (33)
where z f t −ωζkif t + ε f t is the structural error that contains the measurement error in intangible
capital. It is immediately apparent that any control function approach that do not take intoaccount this unobserved measurement error would bias the estimates of output elasticity ofintangible capital. Drawing insights from Collard-Wexler and De Loecker (2016) and combin-ing them with our baseline results, we can argue that the presence of measurement error inintangible capital would downward bias its output elasticities. Therefore, our initial claim thatintangible capital is an important input in production still holds and our estimates of its inputshare are lower bounds. We leave the precise estimation that is robust to measurement errorfor future work.
Sector-Level Production Functions: Here, instead of assuming that each firm produceswith the same technology, we assume that firms share the same technology only within sectors(2 Digit NAICS). Among others, this allows for sector-time dependent price of intangible in-vestment. The input elasticities are aggregate with sector sales as weights. Figure B.10 showsthe evolution of the intangible capital input share and the evolution of the labor share. Bothshow a very similar patter with our main specification, i.e., the share of intangible capital in-creases over the periods and the share of labor decreases.50 This reinforces our finding thatthe IBTC is a transformation in the firm-level production technology where the labor input issubstituted with the intangible capital input.
50The physical capital share does not show any obvious trend.
56
Figure B.10: Trends in Input shares: Sector-level
0
1
2
3
Inta
ngib
le In
put s
hare
1980 1985 1990 1995 2000 2005Year
(a) Intangible Capital Input Share
.96
.98
1
1.02
1.04
1.06
Labo
r Inp
ut s
hare
1980 1985 1990 1995 2000 2005Year
(b) Labor Input Share
.7
.8
.9
1
1.1
Phys
ical
Cap
ital I
nput
sha
re
1980 1985 1990 1995 2000 2005Year
(c) Physical Capital Input Share
Note. Sectoral Production Function Estimates.
Translog Production Function: Finally, here we assume that firms instead of using a Cobb-Douglass production function they use a Translog production function given by:
log(y f t) = α1 log(kp, f t) + α2 log(ki, f t) + α3 log(` f t)
+ α4 log(kp, f t)2 + α5 log(ki, f t)
2 + α6 log(` f t)2
+ α7 log(kp, f t) log(ki, f t) + α8 log log(kp, f t) log(` f t) + α9 log(ki, f t) log(` f t).
Since the input elasticities are firm-specific, we plot in Figure B.11 the average elasticityacross firms and time. Even when we assume that firms produce with a Translog productionfunction we find that the input elasticity of intangible capital is rising.
57
Figure B.11: Trends in Input shares: Translog Production Function
1
1.2
1.4
1.6
1.8
Inta
ngib
le In
put s
hare
1980 1985 1990 1995 2000 2005Year
(a) Intangible Capital Input Share
.9
.95
1
1.05
1.1
Inta
ngib
le In
put s
hare
1980 1985 1990 1995 2000 2005Year
(b) Labor Input Share
.75
.8
.85
.9
.95
1
Inta
ngib
le In
put s
hare
1980 1985 1990 1995 2000 2005Year
(c) Physical Capital Input Share
Note. Translog Production function estimates.
PFII - Estimation with Alternative Measure of Intangible Capital: Here we re estimatethe production function with all the four empirical IO methods presented so far. Instead ofusing our baseline measure of intangible capital we use the on proposed by Peters and Taylor(2017) and Ewens, Peters, and Wang (2019) (see Appendix A.2). Table B.5 presents the results.The main message remains the same. Intangible capital share in production rises dramaticallyin all the specifications and it does so at the expense of the labor share − the only exceptionin when we use CS, where instead intangible capital share rises at the expense of the physicalcapital share.
58
Table B.5: Production Function Estimates with Alternative Measure of Intangible Capital
(1) (2) (3) (4) (5) (6) (7) (8)
1980-1990 2000-2007
OP DPD CS ACF OP DPD CS ACFlog(y f t) log(y f t) log(y f t) log(y f t) log(y f t) log(y f t) log(y f t) log(y f t)
log(kp, f t) 0.101 0.207 0.174 0.248 0.151 0.271 0.087 0.292(0.033) (0.013) (0.00) (0.080) (0.014) (0.013) (0.00) (0.104)
log(ki, f t) 0.381 0.287 0.105 0.269 0.406 0.378 0.156 0.379(0.041) (0.037) (0.00) (0.147) (0.02) (0.014) (0.00) (0.153)
log(` f t) 0.511 0.525 0.672 0.497 0.456 0.342 0.686 0.316(0.009) (0.006) (0.00) (0.274) (0.009) (0.018) (0.00) (0.33)
Obs. 27,460 21,073 27,460 21,073 20,139 13,814 20,139 13,814
Notes: production function estimates with ACF, CS, OP and DPD. log(kp, f t) is physical capital, log(ki, f t) is intan-gible capital as measured by Peters and Taylor (2017) and Ewens, Peters, and Wang (2019), log(` f t) is the employ-ment, and log(y f t) is value added, i.e., sales minus material costs. Material costs are not reported in Compustatsince they are bundled together with labor costs in the expenditure voice Cost of Goods Sold (see Appendix B).Therefore, to extract material costs we follow Bloom and Van Reenen (2007). Robust standard errors are in paren-theses. We control for year fixed effects for each specification. For the OLS, we also control for firm fixed effects.
PFIII - Estimation with XSGA as and Additional Input: Here we estimate a productionfunction augmented by XSGA as an additional static input in the spirit of De Loecker, Eeck-hout, and Unger (2020). The production function is given by:
log(y f t) = α log(kp, f t) + ω log(ki, f t) + ν log(` f t) + ξ log(SGA f t) + zit + ε f t. (34)
Table B.6 shows the results. As we can see adding an additional input does change the levelestimates significantly but steel the intangible shows a significant rise in all the specifications.
59
Table B.6: Production Function Estimates with XSGA as an Additional Input
(1) (2) (3) (5) (6) (7)
1980-1990 2000-2007
OP CS ACF OP CS ACFlog(y f t) log(y f t) log(y f t) log(y f t) log(y f t) log(y f t)
log(kp, f t) 0.144 0.08 0.04 0.214 0.046 0.228(0.003) (0.00) (0.15) (0.017) (0.00) (0.02)
log(ki, f t) -0.00 0.012 0.018 0.013 0.041 0.033(0.008) (0.00) (0.006) (0.006) (0.00) (0.015)
log(SGA f t) 0.483 0.466 0.423 0.335 0.427 0.452(0.014) (0.00) (0.026) (0.009) (0.00) (0.051)
log(` f t) 0.406 0.377 0.392 0.644 0.403 0.322(0.014) (0.00) (0.051) (0.009) (0.00) (0.110)
Obs. 19,692 19,692 15,035 17,366 17,366 11,761
Notes: Production function estimates with OP, CS, and ACF. log(kp, f t) is physical capital, log(ki, f t) is intangiblecapital, log(SGA f t) is XSGA (a proxy for overhead costs from De Loecker, Eeckhout, and Unger, 2020) log(` f t) isthe employment, and log(y f t) is value-added. Material costs are not reported in Compustat since they are bundledtogether with labor costs in the expenditure voice Cost of Goods Sold (see Appendix B). Therefore, to extractmaterial costs we follow Bloom and Van Reenen (2007). Robust standard errors are in parentheses. We control foryear fixed effects for each specification.
PFIV - Estimation with COGS as the Variable Input: We estimate here a production func-tion whose variable input is COGS. We do so since this has been the main specification usedin De Loecker, Eeckhout, and Unger (2020). The production function is given by:
log(y f t) = α log(kp, f t) + ω log(ki, f t) + ν log(COGS f t) + zit + ε f t. (35)
Despite the fact that for them was a necessity to use COGS, due to their estimation proce-dure of markups, this implicitly generates a misspecification problem in the production func-tion. Infact, COGS = `+ m, where m is material inputs. Therefore, using COGS as variableinput implies that there is full substitutability between labor and materials in production. Wedo not use in this specification the ACF estimator since this method is identified only whenvalue added is used as a measure of firm-level output. Table B.7 presents the results. It can beseen that the estimates reported confirm the results presented throughout the paper.
60
Table B.7: Production Function Estimates with COGS
(1) (2) (3) (4)
1980-1990 2000-2007
OP CS OP CSlog(y f t) log(y f t) log(y f t) log(y f t)
log(kp, f t) 0.118 0.061 0.135 0.064(0.011) (0.074) (0.007) (0.087)
log(ki, f t) 0.030 0.020 0.075 0.076(0.002) (0.047) (0.007) (0.113)
log(COGS f t) 0.828 0.918 0.766 0.860(0.007) (0.093) (0.000) (0.144)
Obs. 19,989 19,989 17,366 17,855
Notes: Production function estimates with OP, CS. log(kp, f t) is physical capital, log(ki, f t) is intangible capital,log(SGA f t) is XSGA (a proxy for overhead costs from De Loecker, Eeckhout, and Unger, 2020) log(COGS f t) is thevariable costs expenditure, and log(y f t) is sales. Robust standard errors are in parentheses. We control for yearfixed effects for each specification.
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C Appendix: Investment Rate Distributions
In this section of the Appendix we present some robustness analysis over the patterns pre-sented on the investment rate distribution of intangible capital investment.
C.1 Investment Rate Distribution across Sectors and Time
First, we document that the patterns presented in the main body of the paper is clear in all themain sectors of the US economy. For this reason we present in table C.8 the main moments ofthe investment distribution of intangible capital for Manufacturing (SIC 20), Wholesale (SIC50), Retail (SIC 52) and Services (SIC 70). All the insights we got in section 3.2 go through here.Particularly, we can observe very high inaction rates in all the sectors. This shows that thetechnological properties of this capital are not a between sector artifact but something deeplyrooted at firm-level. Moreover, we notice that very few negative investments is also very ro-bust fact across sector, suggesting that what we capture is indeed a real economic phenomenon.The only measure that suggests some significant heterogeneity is the spikes rate, although wenotice that this is not a very well define concept in the presence of intangible capital invest-ment since this capital has a very high mean investment rate, i.e. meaning that sometimes theaverage investment is capture as a spike.51
Table C.8: Moments of Investment Rate distribution: By Sector
Investment rates Manufacturing Wholesale Retail Services
Average 0.38 0.21 0.21 0.33Positive fraction, i > 1 0.92 0.82 0.87 0.91Negative fraction, i < −1 0.01 0.04 0.04 0.03
Inaction rate 0.07 0.14 0.09 0.06
Spike rate, |i| > 20 0.80 0.52 0.44 0.68Positive spikes, i > 20 0.80 0.51 0.43 0.67Negative spikes, i < −20 0.01 0.02 0.01 0.01
Standard Deviation 0.25 0.21 0.19 0.27Serial correlation, Corr(it, it−1) 0.32 0.10 0.22 0.31
Note. Moments from the intangible investment rate distribution of Manufacturing (SIC 20), Wholesale (SIC 50), Retail (SIC 52)and Services (SIC 70). For each sector we use a balanced panel of firms between the years 1980 and 1990.
Finally, we document that the investment rate distribution does not display significantlydifferent moments over time. This is shown in table C.9. All the main moment of interestpresent the same strong patterns of underling lumpiness, e.g., inaction rates are significantlyhigher in all sample periods relative to physical capital. Moreover, the investment rate distri-bution does not show any significant measure of negative investment.
51We report spike rates to make our work comparable to the previous literature. However, aware of the lack ofsignificance of this measure, we never use it as an identifying moment in our analysis.
62
Table C.9: Moments of Investment Rate distribution: By Time-Period
Investment rates 1981 –1990 1991 – 2000 2001–2010
Average 0.36 0.35 30Positive fraction, i > 1 0.90 0.89 0.91Negative fraction, i < −1 0.02 0.03 0.04
Inaction rate 0.08 0.09 0.05
Spike rate, |i| > 20 0.75 0.71 0.71Positive spikes, i > 20 0.74 0.70 0.69Negative spikes, i < −20 0.01 0.01 0.02
Standard Deviation 0.26 0.29 0.26Serial correlation, Corr(it, it−1) 0.27 0.23 0.14
Finally, we plot in Figure C.12 how the overall investment rate distributions for both phys-ical and intangible capital look like if we do not take a balanced panel. The figure showspatterns that are totally consistent with our benchmark analysis. This is not completely sur-prising since it is well known that selection dynamics in Compustat are very different from theone in the real economy.
Figure C.12: Investment Rate Distributions
0
.05
.1
.15
.2
-.5 0 .5 1
(a) Intangible Capital
0
.05
.1
.15
.2
-.5 0 .5 1
(b) Physical Capital
Note. Investment Rate distribution for FULL panel of firms between the years 1980 and 1990. To construct these histograms wedrop all the observation from the balanced panel - constructed in the Compustat data as explained in the text - above 2 and allthe observation below -1. Results are robust to other winsorization schemes.
This robustness suggests that the underling technological constraints behind these lumpyinvestment rate distributions are not sector-specific nor time-specific. This gives us confi-dence in assuming that the underling technological properties of intangible capital are com-mon across sector and fixed across time.
63
C.2 Skewness and Kurtosis
The skewness measures the degree and direction of asymmetry. A normal distribution hasa skewness of 0, whereas, a distribution that is skewed to the left has a negative skewness.The kurtosis instead is a measure of the heaviness of the tails of a distribution. A normaldistribution has a kurtosis of 3. We compute skewness and kurtosis and document that bothdistributions are positively skewed and have a kurtosis that is far greater than 3 (see TableC.10). Furthermore, we also compute skewness and kurtosis for the investment rate distribu-tion of intangible capital without zeros. Again, the normality is rejected as the distribution isright skewed and have thick tails.
Table C.10: Investment Rate Distributions: Skewness and Kurtosis
Investment rates physical Intangible Intangible (no zeros)
Skewness 3.74 0.99 1.15Kurtosis 33.82 7.46 8.84
64
D Appendix: Structural Analysis
Table D.11: Parameters Values in General Equilibrium
Parameter Values 1980 Values 2010 Description
Fixed:R 1.05 1.05 Annual interest rateδp 0.07 0.07 Annual depreciation rate physical capitalδi 0.28 0.28 Annual depreciation rate intangible capitalM 0.03 0.03 Measure of prospective entrantsα 0.24 0.255 Physical capital shareφ 0.03 0.095 Intangible capital shareν 0.67 0.595 Labor shareρz 0.85 0.85 Autocorrelation idiosyncratic productivityσz 0.20 0.20 Standard deviation idiosyncratic productivity
Fitted:γp 0.037 0.037 Convex adjustment cost physical capitalγi 0.165 0.165 Convex adjustment cost intangible capitalfp 7e-5 7e-5 Fixed adjustment cost physical capitalfi 4.4e-3 4.4e-3 Fixed adjustment cost intangible capitalce 0.11 0.11 Entry costc f 2.045 2.045 Operating costη 3.05 3.05 Pareto distribution
Table D.12: Untargeted Moments
Model Data
Intangible:Mean investment 0.36 0.36Positive investment 0.70 0.90Positive spike rate 0.68 0.74Negative investment 0.19 0.02
Physical:Mean investment 0.15 0.12Positive investment 0.68 0.90Positive spike rate 0.30 0.14Negative investment 0.29 0.07
The model performs well on non-targeted moments such as mean investment rate, positiveand negative investment rate, and positive spike rate. We list them in table D.12 in AppendixA. In line with data, the model identifies higher spikes in the investment rate distribution rela-tive to that of physical capital. The bi-modality of the investment rate distribution of intangiblecapital is a natural byproduct of the model.
65
D.1 Identification of the productivity process
In our baseline specification, the persistence of the idiosyncratic process is ρz = 0.85 and thestandard deviation is σz = 0.20 (see Foster, Haltiwanger, and Syverson, 2008 and Clementi andPalazzo, 2016a). We do not use the production function estimation to estimate the values ofTFP process because our sample contains only large firms and thus, it is unsuitable to use thatprocess to match the overall cross-section of firms in the economy. However, this assumption isimportant as far as the estimation of adjustment costs are concerned. As discussed in Clementiand Palazzo (2016a), the four moments of the investment rate distribution (i.e., mean, standarddeviation, auto-correlation and inaction rate) do not uniquely identify a vector of parametervalues for adjustment cost. These parameters crucially depends on the TFP process.
As a possible way out, Clementi and Palazzo (2016a) suggest to add another theoreticalrestriction in order to uniquely identify the parameters of adjustment costs (i.e., to match un-conditional standard deviation of the capital–output ratio, to pin down standard deviation ofthe TFP process). However, capital–output ratio is the inverse of the ARPKp. In our mainspecification, we do not match the dispersion in ARPKp but still its values were very similarin the model and the data (i.e., 0.85 in the model and 0.8 in the data). This shows that theparameters of adjustment costs in baseline
D.2 Quasi-Fixed Inputs
First, we plot the distributions of ARPKi and ARPKp in the model and the data. As showsin Figure D.13, the model does a good job in matching the overall distribution of the averageproducts.
The intensive margin of firm investment in intangible capital is described by the followingfirst order condition,
ω Eε ARPKi,t+1 = r + δi + R∂ACi,t
∂kt+1+
∂ACi,t+1
∂ki,t+1
where, ARPKi,t+1 is the average revenue product of intangible capital. ACi,t represent adjust-ment costs associated with intangible capital at time t. The first order condition tells us thatthe ARPKi,t+1 in this economy will be dispersed across firms. This dispersion is caused bytwo forces: the uncertainty in the productivity shock, and the presence of adjustment costs.Given the fact that adjustment of intangible capital is costly, firms do not equalize the expectedARPKi,t+1 with the rental rate of capital r + δi.
Eε MRPKt+1 = r + δ
We plot the dispersion in ARPKi against the dispersion in ARPKp for each industry at SIC3level and document that sd(ARPKi) is higher than sd(ARPKp) in almost all the industries (seeFigure D.14).
Furthermore, we provide additional evidence to corroborate our findings on the dispersionin average products. particular, it is possible that part of the dispersion in the average productof intangible capital as computed in the data is driven by measurement error. To overcome
66
Figure D.13: Dispersion in Average Products
-6 -4 -2 0 2 40
0.1
0.2
0.3
0.4
De
nsity
arpki
arpkp
(a) Model-5 0 5
0
0.1
0.2
0.3
0.4
0.5
0.6
De
nsity
(b) Data
Note.
Figure D.14: Sectoral Dispersion in ARPKp and ARPKi
0
1
2
3
4
sd(A
RPK p
)
0 1 2 3 4sd(ARPKi)
SIC3-Sectors
Note. It plots the standard deviation of ARPKp on the y-axis and standard deviation of ARPKi , on the x-axis, averaged over allyears in the sample period. Marginal products are measured using equation 17. Each circle represents a SIC3 sector, where sizeof the circle is proportional to its size (sale-weighted) in Compustat.
these concerns, we run the following regression specification:
ARPKj, f t+1 = γ0 + γ1ε it + γ2k j, f t + γ3zit−1 + γs + γt + ε f t, j ∈ p, i, (36)
where the dependent variable ARPKj, f t is the average product of input k j and ε it equals zit −zit−1, which we interpret from the lens of the model as a structural exogenous innovation tofirm productivity. We control for firm-level input k j at time t and lagged productivity zit−1 tosee how the average products of two comparable firms change when they are hit by heteroge-neous productivity shocks. Finally, γs and γt are sector and year fixed effects, respectively, andγ1, which is the coefficient of interest, captures how average product respond to a productivityshock.
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ARPKj, f t = γ0 + γ1ε f tγ f + γt + ε f t, j ∈ p, i, (37)
Furthermore, by using the within firm variation in the productivity shocks, we show thatthe average product of intangible capital reacts more to the shocks than the average productof physical capital at the firm level (see Table D.14).
Table D.13: Heterogeneous Response of Average products to TFPR shocks
(1) (2)
Dependent Variable ARPKp, f t ARPKi, f t
ε it 1.243 1.492(0.010) (0.020)
kp, f t -0.160(0.001)
TFPR f i,t−t, 1.165 1.286(0.005) (0.011)
ki, f t -0.446(0.002)
Constant -7.902 -6.962(0.044) (0.087)
Time dummies X XSector dummies X XObservations 53,089 53,089R-squared 0.718 0.686
Notes. We report the coefficient of the regression ARPKp, f t and ARPKi, f t against productivity shock εit. Thecontrols include TFPRit−1, lagged productivity, kp, f t, physical capital, and ki, f t, intangible capital. We restrict thesample period between 1980-2007. Standard errors are in the parentheses. All coefficients are significant at 1%level.
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Table D.14: Heterogeneous Response of Average products to TFPR shocks
(1) (2) (3) (4)
Sample Period 1980-1990 1980-2007
Dependent Variable ARPKp, f t ARPKi, f t ARPKp, f t ARPKi, f t
ε it 0.580 0.802 0.560 0.75(0.012) (0.029) (0.007) (0.016)
Constant 0.862 3.064 1.062 2.420(0.002) (0.005) (0.002) (0.003)
Time dummies X X X XFirm dummies X X X XObservations 20,695 20,695 59,255 59,255R-squared 0.910 0.864 0.860 0.771
Notes. We report the coefficient of the regression ARPKp, f t and ARPKi, f t against productivity shock εit. All coeffi-cients are significant at 1% level.
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D.3 IBTC and Macroeconomic Implications
Table D.15: Intangible Share and Firm-Dynamics
Physical capital share α = 0.24 α = 0.24 α = 0.24 α = 0.24Intangible share ω = 0.03 ω = 0.06 ω = 0.09 ω = 0.10labor share ν = 0.67 ν = 0.64 ν = 0.61 ν = 0.60
Wage 100 93 87 85Employment 100 86 77 72Measure of firms 100 84 67 62ki/l (mean) 100 153 200 223xi/y (mean) 100 135 209 257TFPR (sd) 100 102 105 109TFPR (mean) 100 104 107 108Profit rate (weighted mean) 100 112 125 130HHI 100 117 122 138Average firm size (mean) 20.4 21.1 23.4 23.8
Notes. ki/l is defined as the stock of intangible capital over labor. xi/y is intangible investments over output. Theprofit rate is the sales-weighted average.
In what follows, we provide empirical evidence for the main predictions discussed above.In doing so, we use U.S. Compustat data between 1981 and 1990. We define sectors at SIC2level, and we compute intangible intensity for each sector, along with other variables of in-terest such as concentration, dispersion in TFPR, sales-weighted profit rates and intangibleinvestment over output. The measurement methodology of these variables is explained in sec-tion 2.1. We present the scatter plots for these variables against intangible intensity in Figure 9(see Table D.16 in for corresponding regressions with time and sector fixed effects).
Figure D.15: HHI with Employment
Slope = 0.0011
.16
.17
.18
.19
.2
HH
I (e
mp
loye
me
nt)
0 5 10 15
Intangible Intensity
Note. Binscatter plot with intangible intensity (i.e., stock of intangibles divided by employment) on x-axis and ony-axis. Each blue dot represent mean of the x-axis and y-axis variables within equally-sized bin. The slopes of thelinear-fit provided in each plot is computed while controlling for year and sector fixed effects. The y-axes reportthe intangible intensity of the sector, defined by the ratio of intangibles to physical capital, at SIC2 level.
Similar to what is predicted by the model, we find that firms in intangible-intensive sec-
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Table D.16: Intangible Intensity and Sectoral Implications: SIC2
(1) (2) (3) (4) (5) (6) (7) (8)VARIABLES xi/y HHI (sales) HHI (Emp) π/y sd(TFPR) mean(TFPR) Mean Emp Mean Sales
Intangible Intensity 0.00167*** 0.00198** 0.00112 0.00118** 0.00524*** 0.00562*** 5.63e-05 73.16**(0.000) (0.001) (0.001) (0.000) (0.002) (0.002) (0.000) (36.526)
Constant 0.00214* 0.166*** 0.164*** 0.0380*** 0.320*** 8.289*** 0.00785*** 1,322***(0.001) (0.007) (0.008) (0.004) (0.012) (0.015) (0.001) (270.451)
Observations 584 585 585 585 583 585 585 585R-squared 0.850 0.916 0.908 0.738 0.551 0.886 0.905 0.912Time FE X X X X X X X XSIC2 FE X X X X X X X X
Notes: xi/y is intangible investment over sales and π/y sales-weighted profit rate. Robust standard errors inparentheses. *** p<0.01, ** p<0.05, * p<0.1
Table D.17: Intangible Intensity and Sectoral Implications: SIC3
(1) (2) (3) (4) (5) (6) (7)VARIABLES xi/y HHI (sales) HHI (Emp) π/y sd(TFPR) mean(TFPR) Mean Sales
intangible Intensity 0.00134*** 0.000236 0.000609 -0.000278 0.00257*** 0.00282*** 17.47*(0.000) (0.000) (0.000) (0.000) (0.001) (0.001) (8.938)
Constant 0.00526*** 0.338*** 0.343*** 0.0542*** 0.287*** 8.355*** 1,338***(0.001) (0.004) (0.004) (0.002) (0.006) (0.007) (76.474)
Observations 2,119 2,120 2,120 2,120 2,035 2,097 2,120R-squared 0.796 0.841 0.826 0.680 0.563 0.823 0.943Time FE X X X X X X XSIC3 FE X X X X X X X
Notes: xi/y is intangible investment over sales and π/y sales-weighted profit rate. Robust standard errors inparentheses. *** p<0.01, ** p<0.05, * p<0.1
tors spend more in intangible investments as a percentage of their output. Further, the modelpredictions about increasing (mis)allocation, i.e., increasing dispersion in TFPR, in sectors’ in-tangible intensity is also supported in the data. This finding suggests that a part of the misallo-cation in the intangible-intensive sectors is driven by the technological constraints rather thanby any sort of firm-level distortion. Also, similar to what we document in the model, meanTFPR is also rising with intangible intensity in the data as well. Finally, we document that theintangible-intensive sectors have higher sales-weighted profit rate and higher concentration asmeasured by HHI as well (see Figure Figure D.15 for employment based HHI).
yst = φ0 + φ1 I Ist + φs + φt + εst (38)
D.3.1 Discussion: Sales Shares and Concentration
Our baseline measure of concentration is sales based Herfindahl-Hirschman Index (HHI).Here, we also compute other concentration measure such as the fraction of sales accrued by thethirty largest firms (CR-30), the fraction of sales accrued by the twenty largest firms (CR-20)and the fraction of sales accrued by the four largest firms (CR-4). The CR-20 and CR-4 havebeen used in Autor, Dorn, Katz, Patterson, and Van Reenen (2017). In our framework, theconcentration of economic activity increases with IBTC. As shows in figure D.17, the increasein concentration is driven by a decline in the proportion of low sales-share firms relative to
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Figure D.16: Dispersion in TFPR over time
-1 -0.5 0 0.5 10
0.5
1
1.5
De
nsity
1980-1990
2000-2007
(a) Distribution of TFPR - Model-2 -1 0 1 2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
De
nsity
(b) Distribution of TFPR - Data
high sales-share firms. The increase in HHI is consistent with the increase in CR-4, CR-20 andCR-30.
Figure D.17: Sale Shares and Concentration in the Model
0 2 4 6 8
Sales Share 10-4
0
0.2
0.4
0.6
Sh
are
of
Firm
s
1980
2007
(a) Distribution of Sale Shares
0 0.05 0.1 0.15 0.2 0.25
% Changes between 1980-2007
HHI
CR-30
CR-20
CR-4
Sa
les B
ase
d
(b) Concentration
Note. We compute sales-based Herfindahl-Hirschman Index (HHI), the fraction of sales accrued by the thirty largest firms (CR-30), the fraction of sales accrued by the twenty largest firms (CR-20) and the fraction of sales accrued by the four largest firms(CR-4) in the model.
D.4 Alternative Mechanisms
Here in this section of the Appendix we inspect alternative mechanisms that have been em-phasized by the literature such as: The rise of fixed costs, the decline in the prices of intangiblecapital relative to physical capital, and the decline in the risk-free interest rate.
D.4.1 The Rise of Fixed costs
A number of papers have argued that the rise of fixed costs in production is responsible forthe underlying changes that the US economy has witnessed in the past three decades (e.g., seeDe Ridder, 2019 and De Loecker, Eeckhout, and Mongey, 2019). In this section, we provide
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results from a counterfactual exercise that studies the effects of rising fixed costs within ourframework. In particular, we discuss the main differences between its implications and theimplications of IBTC.
In order to highlight the macroeconomic effects of the rising fixed costs, we increase c f ,while keeping everything else constant. The value of parameters are provided in Table D.18.The values of the parameters in the benchmark economy (i.e., steady state in 1980s) is keptsame as the one used in the baseline results. In order to calibrate the value of c f in the 2007steady state, we match the change in the ratio of fixed cost to output that we observe in the databetween 1980 and 2007. In Compustat, fixed costs (overhead costs) are measured by XSGA (seesection A for more details). The ratio of total fixed costs to total output has increased by 45%approximately. We estimate c f = 3.26 in 2007 steady state, that gives us a rise of approximately45% in total fixed costs to total output in the model.
Table D.18: Parameters Values in General Equilibrium
Parameter Values 1980 Values 2007 Description
Fixed:R 1.05 1.05 Annual interest rateδp 0.07 0.07 Annual depreciation rate physical capitalδi 0.28 0.28 Annual depreciation rate intangible capitalM 0.03 0.03 Measure of prospective entrantsα 0.24 0.24 Physical capital shareφ 0.03 0.03 Intangible capital shareν 0.67 0.67 Labor shareρz 0.85 0.85 Autocorrelation idiosyncratic productivityσz 0.20 0.20 Standard deviation idiosyncratic productivity
Fitted:γp 0.037 0.037 Convex adjustment cost physical capitalγi 0.165 0.165 Convex adjustment cost intangible capitalfp 7e-5 7e-5 Fixed adjustment cost physical capitalfi 4.4e-3 4.4e-3 Fixed adjustment cost intangible capitalce 0.11 0.11 Entry costc f 2.045 3.230 Operating costη 3.05 3.05 Pareto distribution
We report the main results in Table D.19. The intuition behind the results is straightfor-ward. As fixed costs increase, the productivity threshold of the marginal entrant increases,which further implies that only very productive firms are able to operate profitably in this en-vironment, consequently increasing the average productivity and as well as average size (i.e.,number of employees) of incumbent firms. In other words, the selection effect is very strongin the new steady state. In equilibrium, this implies a reduction in the mass of entrants andas well as the mass of active firms, increasing industry concentration as measured by the HHIindex. Moreover, the sales-weighted profit rates increase by 5 percent.
However, contrary to our baseline results, the new steady state shows a counterfactualdecline in firms’ average intangible capital invest as a share of their output. Furthermore, alsothe level of (mis)allocation of resources in the economy remains unchanged in the new steady
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state. Taking stocks, the rise in fixed costs, while explaining some of the trends in the USeconomy, fails to explain the main objects of interest of this paper such as the rise in intangiblecapital and the rise in (mis)allocation.
Table D.19: Results- Steady State Comparison
% ∆ 1980-2007
1980 S.S. 2007 S.S 2007 S.S Model BDS CompustatIBTC Fixed Costs Fixed Costs
Average firm size 20.5 22 34.2 +67 +8.2 +36Average entrant size 5.6 8.1 14.1 +150 +11 -xi/y 0.03 0.08 0.02 −33 - +460Profit rate 0.18 0.23 0.19 +5.5 - +37sd(TFPR) 0.24 0.26 0.24 +0 - +22HHI 7.7e-4 9.7e-4 1.1e-3 +42 - +39
Notes. We report average size of firms and average size of incumbents with Business Dynamics statistics (BDS)data available online. For all the statistics reported for Compustat data, except for profit rate, we compute sectoraverages at SIC 2-digit and then weight it by sector sales share to compute yearly averages. For profit rate, wecompute sales-weighted average at SIC 2 digit level and then weight it by sector sales share to compute yearlyaverages.
D.4.2 The Secular Decline in the Relative Price of Intangible Investments
In the baseline model, we assume that the relative price of intangible investments, pi, is 1,meaning that investing in intangible capital is as costly as investing in physical capital. More-over, we assumed this relation to be constant over time. However, as emphasized by Zhang(2019) , we know that the relative price of intangible capital has declined steadily over theyears.
In this section, we will study the implications of such a decline and we will compare themwith the implications implied by the IBTC. To do so we have to introduce into the model therelative price of investing in intangible capital. For the sake of illustration we will show whatchanges in the model only for the case of a firm which invests in both capital. But the samelogic carries on for all the value functions presented in Section 4. The value function of a firminvesting in both capitals becomes:
V1(z, kp, ki; w) = maxk′p,k′i−pixp − xi − C(xp, xi; kp, ki) +
1R
∫V(z′, k′p, k′i; w)Γ(dz′|z),
s.t. k′p = (1− δp)kp + xp,
k′i = (1− δi)ki + xi.
(39)
In order to see what are the implications of the model when the relative price of intangiblecapital declines we recalibrate the benchmark economy, i.e., the steady state economy in the1980s, keeping all the parameter fixed except for pi which goes from 1 to 0.66. Such a 44%decline is consistent with what we observe in the data (see Zhang, 2019). Table D.20 shows theparametrization in the two economies.
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Table D.20: Parameters Values in General Equilibrium
Parameter Values 1980 Values 2007 Description
Fixed:R 1.05 1.05 Annual interest rateδp 0.07 0.07 Annual depreciation rate physical capitalδi 0.28 0.28 Annual depreciation rate intangible capitalM 0.03 0.03 Measure of prospective entrantsα 0.24 0.255 Physical capital shareφ 0.03 0.095 Intangible capital shareν 0.67 0.595 Labor shareρz 0.85 0.85 Autocorrelation idiosyncratic productivityσz 0.20 0.20 Standard deviation idiosyncratic productivitypi 1.00 0.66 Price of Intangible Investments
Fitted:γp 0.037 0.037 Convex adjustment cost physical capitalγi 0.165 0.165 Convex adjustment cost intangible capitalfp 7e-5 7e-5 Fixed adjustment cost physical capitalfi 4.4e-3 4.4e-3 Fixed adjustment cost intangible capitalce 0.11 0.11 Entry costc f 2.045 2.045 Operating costη 3.05 3.05 Pareto distribution
Results are shown in Table D.21. In our model, the decline in the relative price of intangi-ble capital fails to explain all the trends we observe in the US economy. In this scenario firmsbecome smaller and invest less. The average profit in the economy declines due to a relaxationin the selection process. Moreover, both concentration and (mis)allocation decline. We con-clude that this channel, relative to IBTC, has counterfactual implications for the main objectsof interest of this paper.
Table D.21: Results- Steady State Comparison
% ∆ 1980-2007
1980 S.S. 2007 S.S 2007 S.S Model BDS CompustatIBTC Pi/Pp = 0.66
Average firm size 20.5 22 19.3 −5 +8.2 +36Average entrant size 5.6 8.1 5.2 −7 +11 -xi/y 0.03 0.08 0.04 +33 - +460Profit rate 0.18 0.23 0.17 −5 - +37sd(TFPR) 0.24 0.26 0.24 +0 - +22HHI 7.7e-4 9.7e-4 6.8e-4 −11 - +39
Notes. We report average size of firms and average size of incumbents with Business Dynamics statistics (BDS)data available online. For all the statistics reported for Compustat data, except for profit rate, we compute sectoraverages at SIC 2-digit and then weight it by sector sales share to compute yearly averages. For profit rate, wecompute sales-weighted average at SIC 2 digit level and then weight it by sector sales share to compute yearlyaverages.
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D.4.3 The Secular Decline in the Risk-free Interest rate
In this section, we discuss the macroeconomic implications of the secular decline in the risk-free interest rate in our model and how it affects our baseline results. This channel has beenfirst put forward by Liu, Mian, and Sufi (2019). In our model, the risk-free interest rate isdirectly calibrate. Therefore, in order to compare the implications of such a decline with theimplications given by IBTC we just recalibrate the risk-free interest rate from 5% to 2% keepingall the other parameter fixed. This change is consistent with measurement by Rachel and Smith(2015). Table D.22 shows our calibration.
Table D.22: Parameters Values in General Equilibrium
Parameter Values 1980 Values 2007 Description
Fixed:R 1.05 1.02 Annual interest rateδp 0.07 0.07 Annual depreciation rate physical capitalδi 0.28 0.28 Annual depreciation rate intangible capitalM 0.03 0.03 Measure of prospective entrantsα 0.24 0.24 Physical capital shareφ 0.03 0.03 Intangible capital shareν 0.67 0.67 Labor shareρz 0.85 0.85 Autocorrelation idiosyncratic productivityσz 0.20 0.20 Standard deviation idiosyncratic productivity
Fitted:γp 0.037 0.037 Convex adjustment cost physical capitalγi 0.165 0.165 Convex adjustment cost intangible capitalfp 7e-5 7e-5 Fixed adjustment cost physical capitalfi 4.4e-3 4.4e-3 Fixed adjustment cost intangible capitalce 0.11 0.11 Entry costc f 2.045 2.045 Operating costη 3.05 3.05 Pareto distribution
Results are shown in Table D.21. In our model, the decline in the risk-free interest ratefails to explain all the trends we observe in the US economy. In this scenario firms becomesmaller and invest less. The average profit in the economy declines due to a relaxation in theselection process. Moreover, both concentration and (mis)allocation decline. We conclude thatthis channel, relative to IBTC, has counterfactual implications for the main objects of interestof this paper.
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Table D.23: Results- Steady State Comparison
% ∆ 1980-2007
1980 S.S. 2007 S.S 2007 S.S Model BDS CompustatIBTC Decline in R
Average firm size 20.5 22 13 −36 +8.2 +36Average entrant size 5.6 8.1 3 −46 +11 -xi/y 0.03 0.08 0.02 −33 - +460Profit rate 0.18 0.23 0.10 −44 - +37sd(TFPR) 0.24 0.26 0.25 +4 - +22HHI 7.7e-4 9.7e-4 4.2e-4 −45 - +39
Notes. We report average size of firms and average size of incumbents with Business Dynamics statistics (BDS)data available online. For all the statistics reported for Compustat data, except for profit rate, we compute sectoraverages at SIC 2-digit and then weight it by sector sales share to compute yearly averages. For profit rate, wecompute sales-weighted average at SIC 2 digit level and then weight it by sector sales share to compute yearlyaverages.
D.5 Intangible Capital, Human Capital and Identification of Adjustment costs
In principle, conceptually, it is difficult to disentangle intangible capital and human capital(see Eisfeldt, Falato, and Xiaolan, 2019, and Koh, Santaeulalia-Llopis, and Zheng, 2016). Inour framework and estimation, a part of human capital is absorbed in our measure of intangi-ble capital, which is a broader concept. Therefore, the estimated adjustment costs of intangi-ble capital include, for instance, the search costs that firms have to pay for high skilled laborsearch. This reading of the results is consistent with the literature that argues that the creationof intangible capital requires skilled human capital which is costly to adjust.
Further, the traditional labor share (wage bill) could be mismeasured (it is one of the argu-ments in Eisfeldt, Falato, and Xiaolan, 2019, and Koh, Santaeulalia-Llopis, and Zheng, 2016).In principle, it is possible that the way firms pay to high skilled labor is by sharing the returnson intangible capital that they jointly created. One way to interpret the declining labor share isthat there exist different types of labor: low skilled and high skilled. The share of low skilledlabor is actually declining. The share of high skilled labor is partly captured by our measure ofintangible investments. For example, in a model, where firms and labor together create intan-gible capital (for instance, training of workforce in an environment of limited commitment),firms would sign long term contracts with the high skilled employees (see Sun and Xiaolan,2019, and Döttling, Perotti, and Ladika, 2019). These investments do entail high adjustmentcosts. Therefore, our framework (parsimonious in its way of modeling) captures better thenature of firms investment process and production.
Finally, alternatively to our hypothesis, the secular rise in the supply of high skilled labormay have incentivized firms to shift their production process towards inputs that complementhigh skilled labor, intangible capital. We want to emphasize on the point that, we do nottake a stand on the causes of IBTC.52 The results from the production function estimationremains valid and unbiased. The baseline results depend on the finding that intangibles entailshigher adjustment costs and any sort of force that causes the production process to become
52This is out of the scope of this paper, however, we do think that it is important to understand what lies behindIBTC and is a potential avenue for future research.
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more intensive in intangible capital, IBTC, would support our mechanism and therefore, thebaseline results remain valid.
D.6 Financial Frictions, Identification of Adjustment costs and IBTC
In our baseline model, we do no have size dependent distortions such as liquidity constraints.However, we could analyze the effects of financial frictions with the help of the results insection 5. The identification of convex costs in the model is crucially depends on the serialcorrelation of the investment rate. David and Venkateswaran (2019) argue this sort identifica-tion would bias down the estimates of convex costs in an environment where size dependentdistortions (financial frictions being one of them ) are important. The reason being that sizedependent distortions reduce the sensitivity of investment to serially correlated factors andthus, increasing the importance of transitory factors. Therefore, in our baseline estimation,we argue that our way of estimating parameters of convex costs is only measuring the lowerbound. Hence, we are not overestimating the impact of adjustment costs on firm dynamics.
Further, many paper have argued that intangible capital is less collateralizable than phys-ical capital (e.g., see Caggese and Perez-Orive, 2017) and therefore, it is important to analyzehow this attribute of intangible capital would conflate into our baseline results. In Caggeseand Perez-Orive (2017), the interaction between financial friction and indivisibilities in invest-ment dampens the response of the firms to aggregate shocks (e.g., monetary policy shocks).Therefore, our results complements their findings. Moreover, in a framework with adjustmentcosts and financial frictions, IBTC would increase misallocation and concentration, and re-duce firm entry more than our baseline results. Therefore, we argue that while collaterizabilityof intangible capital is an important property, it does not goes against our mechanism ratherits inclusion would imply stronger macroeconomics changes due to IBTC. In contrast, in thepresence of financial frictions, the rise in misallocaiton and decline in firm entry would havenegative welfare implications and the decentralized equilibrium allocation would be differentfrom one that is implied by the social planner.
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