Intangible-Investment-Specific Technical Change, Concentration and Labor Share ∗

Intangible-Investment-Speciﬁc Technical Change, Concentration and Labor Share ∗

Lichen Zhang†

Oct. 18, 2019

JOB MAKET PAPER

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Abstract In the past three decades, measured investment in factors such as software and R&D, collectively known as "intangible capital" has risen relative to the total business income. Together with this trend, the U.S. business sector has also been characterized by increased concentration and declined measured labor share. This paper develops a quantitative GE model of firm dynamics, consistent with important aspects of firms’ behavior at micro level, to explore the connection between the three major trends of the U.S. business sector. Individual firms operate at Cobb-Douglas technology, but they aggregate as if the elasticity of substitution among factors in production is above unity, triggered by an intangible-investment-specific technical change (IISTC). The IISTC shifts the firm distribution towards large, intangible-intensive firms with low labor share. I show that an IISTC, targeted to match the decline in the relative price of intangible investment goods, accounts for more than 2/3 of the rise of concentration and around half of the decline in measured labor share.

Keywords: Intangible Investment, Concentration, Labor Share, Technical Change JEL Classiﬁcation: E13, E22, E23, E25, D21, D25, D33 ∗I’m deeply indebted to my advisors Tim Kehoe, Manuel Amador, Loukas Karabarbounis, and Jonathan Heathcote for their invaluable guidance and continuous support. I’m particularly grateful to Ellen McGrattan for her detailed comments and discussions on this project. I also want to thank Hengjie Ai, David Argente, Anmol Bhandari, Jarda Borovicka, Zhifeng Cai, Doirrean Fitzgerald, Tom Holmes as well as participants in the International Macro/Trade Workshop and Quantitative Macro workshop at the University of Minnesota, and the UMN-UW Macro/International graduate workshop for their useful suggestions and comments. The views expressed herein are those of the author and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System. All errors are my own. †Aﬃliation: University of Minnesota and Federal Reserve Bank of Minneapolis. 1925 Fourth Street South, Minneapolis, MN 55455 (email: [email protected]). 1 Introduction

In the past three decades, measured investment in factors such as software and R&D, collectively known as “intangible capital” has risen relative to the total business income. Together with this trend, the U.S. business sector has been characterized by increased concentration in terms of employment and sales in large firms at the national level as well as declined measured labor share. This paper proposes a theory to explore the connection between the three major trends of the U.S. business sector quantitatively. I show that an intangible-investment specific technical change (IISTC) that drives the intangible investment relative to the total business income can account for the rise of concentration and the declined measured labor share jointly. I highlight the role of intangibles in shaping factor income shares, concentration, and business dynamism at both the micro and aggregate levels. In my model, firms start with zero intangible capital and the same level of wealth in terms of tangible assets, but differ in productivities on producing consumption/physical investment goods and intangible investment goods1. The varied productivity on intangible investment goods leads to heterogeneous factor shares of income across firms in the long run. Firms with high productivity on producing intangibles are large and intangible-capital intensive, and more intangible-capital intensive firms are also less labor-intensive. Hence, my model features a positive correlation between firm size and intangible-intensity, and a negative relationship between firm size and firm-level labor share, which are consistent with the micro evidence2. The IISTC in my model is manifested as a permanent increase in the aggregate productivity on producing intangible investment goods relative to that on producing consumption/physical investment goods, which captures the declined relative price of intangibles, particularly, software, from the data. Although firms operate at Cobb-Douglas technology, they aggregate as if the elasticity of substitution among factors in production is above one, triggered by the IISTC3. Firms with high productivity on producing intangibles are large and intangible-

1Since firms start with the same endogenous conditions and non-exiting firms can eventually grow to their optimal levels, only the exogenous heterogeneity matters in steady states. 2See Autor et al.(2017) and Crouzet and Eberly(2019) for United States and Lashkari, Bauer, and Boussard (2019) for very similar patterns in France. 3Consider a simple example where there are two perfectly competitive firms in each sector: one labor- intensive firm with production technology Y1 = L and the other is a capital-intensive firm with production technology Y2 = AK. There are constant labor shares within each firm. Suppose there is capital-specific technical change such that A increases. Under certain conditions, as capital becomes relatively cheaper, it is

1 intensive, which benefit disproportionately from this intangible-specific technology advances, thus becoming even larger and more intangible-intensive. More specifically, the IISTC pushes up the equilibrium wage rate. Due to this GE effect together with the negative correlation between firm size and firm-level labor share, output and labor are reallocated from small firms with low productivity on producing intangibles to large firms with high productivity on producing intangibles. This leads to the increased concentration. The declined aggregate labor share is the consequence of both a within-firm effect and an across-firm effect. Within each firm, the IISTC makes all the firms in the economy become more intangible-capital-intensive, thus being less labor intensive. Across firms, the IISTC shifts the firm distribution towards large firms with low labor share. My approach is to develop a quantitative GE model of firm dynamics for understanding the aggregate long-run impact of an intangible-investment-specific technical change (IISTC) on concentration and measured labor share. I enrich the model with three key features. First, I explicitly incorporate intangible investment and intangible capital into the model. More specifically, I differentiate between two types capital - tangible and intangible, which are both production inputs but intangible capital is accumulated within each firm through costly production for intangible investment goods given the firm’s production ability4. The goal of doing this is to enable this model to explore the role of intangible investment in shaping factor income shares and concentration at both the micro and macro levels. Second, I introduce financial frictions: incumbent firms cannot issue equity, and they face borrowing constraints that restrict leverage to a multiple of collateralizable assets, as in Evans and Jovanovic(1989). Adding such a financial friction contribute to a more realistic firm life cycle since it hinders the births of start-ups and slows the expansion of young firms. Otherwise, firms jump to their optimal level almost immediately after their births. This implies that statistics related to firm dynamics such as entry and concentration generated from a model without financial friction are going to be very sensitive to technology parameters. Since a main goal of this paper is to study the impact of a technical change on firm dynamics, having a model that features realistic firm distribution and life cycle of firms are very important. Third, I allow for endogenous entry and exit of firms. This helps to understand the effects substituted for labor on the aggregate-level. For more details, see Karabarbounis(2018) 4This differentiates with the existing literature on explaining declined measured labor share such as Aghion et al.(2019) that target "intangible share" without explicitly incorporating both tangible and intangible capital.

2 of an IISTC on how heterogenous firms’ behavior shape the aggregates. In the model, firms with high productivity on intangible investment goods account for a disproportionately large fraction of the total employment and total final output. Moreover, endogenous exit, interacting with financial frictions, makes larger firms harder to exit, which makes firm size and firm age distribution in the model closer to the data. I calibrate the model to the U.S. business sector as if it was at the steady state in early 1980s to match a rich set of both macro and micro moments. In particular, I discipline firms’ production technology to target the BEA-measured income shares and their productivity processes on both consumption/physical investment and intangible investment goods to capture the empirical fact that a small proportion of highly productive firms account for a very large share of total employment and total final output. I then show that the calibrated model can produce firm-level cross-sectional predictions consistent with the micro evidence. The most important one is the negative correlation between firm size and firm-level labor share documented by Autor, Dorn, Katz, Patterson, and Reenen(2017) using census data. As I mentioned above, this relationship is key to generating the declined labor share and the rise of concentration driven by the IISTC. I finally use the calibrated model to quantify the aggregate long-term impact of the IISTC, which is targeted to match the decline in the relative price of intangible-investment goods from early 1980s to 2010s. In particular, I compare two steady states: one is the initial steady state calibrated to early 1980s and the other is the new steady state with the technical change. I find that the IISTC, or fallen relative prices of intangible investment goods, can explain both the decline in the current BEA-measured labor share by around 60%, as well as the the decline in measured labor share of pre-1999 revision when intangibles are not treated as final output by around 40%. Moreover, it accounts for more than two thirds of the increase in the employment share of large firms (with number of employees greater than 500) and of the increase in the share of total sales going to top 10% firms. To inspect the key mechanism that leads to the results above, I explain why the IISTC shifts firm distribution towards large, more intangible-intensive firms with low labor share by deriving the aggregate elasticities of substitution among factors numerically. I verify that the aggregates of production factors - physical and intangible capital, as well as labor are more than Cobb-Douglas substitutes, triggered by the IISTC. Moreover, I consider a number of

3 extensions to the baseline model by altering its key features one by one to understand what are the most essential element of the baseline model and what are the deep parameters driving the key relationship that generate the results above. There are two cautious remarks. First, this paper is not about discussing how should we measure "labor share" but taking the national account measured labor share as given, together with a set of consistent micro evidence, and attempting to provide a coherent story to justify the change 5. As the second cautious remark, my results do not mean, and are far from implying that the IISTC is the only driver of the observed trends for measured intangible investment, measured labor share, and concentration. Compared to papers that have a horse race structure6, this paper attempts to jointly explain a multiple of empirical facts using one driving force. Hence, I abstract my model from many other factors that may potentially affect the results7. The paper is organized as follows. Section 2 is for literature review. Section 3 describes the model set-up and defines a recursive competitive equilibrium. Section 4 discusses calibration and cross-sectional implications. In section 5, I quantify the aggregate long-run effect of the IISTC and inspect the key mechanism that leads to the main results. Section 6 concludes the paper.

2 Related Literature

This paper relates to several strands of literature. First, it contributes to the voluminous literature on the evolution of the labor share in general. Using aggregate-level data, Elsby, Hobijn, and Sahin(2013) document the decline of labor compensation as a share of national

5Koh, Santaeulalia-Llopis, and Zheng(2016) question the way that national accounts measure labor share where the entire intangible capital rents are attributed to capital income rather than labor income. They argue that if part of the intangible investment is financed by workers rationalized by having workers paid wage lower than their marginal value product in return for some equity, then that part of intangible capital rents should be allocated to labor income. However, to which extent should the rents from intangible investment be attributed to labor income is hard to measure. Recently, there are papers on "human capitalist” (see Eisfeldt, Falato, and Xiaolan(2018), Hartman-Glaser, Lustig, and Xiaolan(2019)) that are trying to provide an answer for corporate firms. Bhandari and McGrattan(2019), instead, focuses on entrepreneurs and corrects the labor income by measuring "sweat equity” accumulated through entrepreneurs’ labor hours that are unmeasured by the national account. 6For example, Akcigit and Ates(2019) explores which force plays a dominant role among a multiple of candidates in explaining the slowdown of the business dynamism. 7Typical examples include decline in real interest rate, decline in effective corporate tax rate, higher R&D subsidies, higher entry cost, lower knowledge diffusion, rising markups, change in labor force participation, globalization and trade, among others

4 income for the U.S., and Karabarbounis and Neiman(2014) show that it is also a global phenomenon. Later works such as Autor, Dorn, Katz, Patterson, and Reenen(2017) and Kehrig and Vincent(2018) confirm this trend using micro data from U.S. Census 8. While Karabarbounis and Neiman(2014) emphasize that the decline in labor compensation share both in the U.S. and overseas represents primarily a within-industry rather than a between- industry phenomenon, Autor et al.(2017), by exploring the heterogeneity among firms, find that the fall in the labor share is driven largely by between-firm reallocation rather than a fall in the unweighted mean labor share within firms. The decline of the measured labor share has spurred a growing literature attempting to rationalize it through a wide set of economic mechanisms: increased openness to international trade and outsourcing (Elsby, Hobijn, and Sahin, 2013); an increase in concentration that causes increasing profit rates (Autor et al., 2017; Barkai, 2017), and increasing markups (DeLoecker and Eeckhout, 2017); automation (Acemoglu and Restrepo, 2018), as well as the decline in the relative price of investment good that increases the substitution of capital for labor (Eden and Gaggl, 2018; Karabarbounis and Neiman, 2014). Two recent works (Hubmer, 2018; Karabarbounis and Neiman, 2018) go beyond the period of declining labor share: the former extends that some of the proposed explanations are seemingly at odds with earlier data, and the latter argues that the relatively stable labor share pre-1980s and the declining trend afterwards result from the race between preference working through an income effect that offsets the the capital-labor substitution and an investment-specific technical change that triggers this substitution. I contribute to the literature by linking the firm-level cross-sectional implications from micro data to the macro behavior of labor share. More specifically, I provide a theory built around firms with heterogeneous factor shares of income and discipline the model to match the aggregate income shares and a rich set of firm-level moments so that it can replicate the salient features from micro evidence such as the the negative correlation between firm size and firm-level labor share documented by Autor et al.(2017). Instead of assuming a non-neutral aggregate CES production technology as in most papers (e.g. Acemoglu and Restrepo(2018); Barkai(2017); Eden and Gaggl(2018); Hubmer(2018); Karabarbounis and Neiman(2014)),

8Koh, Santaeulalia-Llopis, and Zheng(2016) argues that the labor share would have declined little if investments into intangible capital were treated as expenditures rather than investments. However, the accounting treatment of intangibles cannot mechanically explain a decline in the payroll-to-sales ratio, or the rising concentration of sales which Autor et al.(2017) ﬁnd to be correlated with declining labor shares at the industry level using Census data.

5 individual firms operate in my model operate at Cobb-Douglas technology, but they aggregate as if the elasticity of substitution among factors in production is above one, triggered by an intangible-investment-specific technical change (IISTC). Consequently, the aggregate labor share declines because the IISTC shifts the firm distribution towards firms with low labor share. Second, my work complements to a set of recent papers concerning increased concentration in the U.S., (Autor et al., 2017; Barkai, 2017; Crouzet and Eberly, 2019; Gutierrez and Philippon, 2017; Hopenhayn, Neira, and Singhania, 2019; Hsieh and Rossi-Hansberg, 2019; Liu, Mian, and Sufi, 2019). They focus on either the increased employment share, or sales share of large firms, or both. Some of them (e.g. Barkai(2017); Gutierrez and Philippon (2017); Liu, Mian, and Sufi(2019)) argue that rising concentration is due to rising mark-ups and declining competition among firms, while others believe that higher market concentration may not necessarily imply higher market power of firms, consistent with Syverson(2004a,b). Autor et al.(2017) and Bessen(2016) argue that rising concentration is a result of higher market competition and the rise of more productive firms. Bessen(2016) documents that sales concentration is strongly correlated with the use of information and communication technologies at the industry level. In a similar vein, Crouzet and Eberly(2019) observe that at the industry level, the firms with the largest size and highest growth rate are the ones whose investment in intangible capital grows the fastest. Hopenhayn, Neira, and Singhania(2019) also has the results that the expansion of most productive firms leads to rising concentration (in terms of employment share), but the driving force is the declining growth rate of labor force participation. Understanding the source of rising concentration can be important since different sources have vastly different welfare and policy implications. If concentration is increasing because of the expansion of the most productive firms, it may be efficient. If, on the other hand, rising concentration reflects greater market power, it may imply inefficiencies and resource misallocation. This paper proposes a theory in line with the latter set of literature: rising market concentration is due to the technical change that favors "superstar firms” rather than rising mark-ups. There are three very recent papers that study labor share and concentration jointly. Au- tor et al.(2017) document that rise in concentration correlates with decline in labor share using U.S. Census data. Akcigit and Ates(2019) quantitatively investigates that the decline

6 in knowledge diffusion is the most dominant force (among a set of candidates) that drives the trends for labor share and concentration simultaneously. Aghion, Bergeaud, Boppart, Klenow, and Li(2019) is most closely related to my work. They argue that reducing firm-level costs of spanning multiple markets, perhaps due to IT advances, leads to rising concentration and declining labor share (driven by change in firm composition). I differentiate my work from theirs in three major aspects: first, I build up a model that explicitly incorporate intangible investment to establish a direct mapping between the measurement and the framework. The driving force is a technical change as a decline in the relative price of intangible investment goods, which can be observed directly from the data. Second, the negative relationship between firm size and firm-level labor share generated from my model that is consistent with micro evidence from Autor et al.(2017) does not depend on the overhead-cost-set-up, which is used by most of the papers able to generate this relationship (e.g. Aghion et al.(2019); Autor et al.(2017); Hopenhayn, Neira, and Singhania(2019)). Instead, in my model, firms are heterogeneous in factor shares of income due to heterogenous productivity on producing intangibles and they operate at Cobb-Douglas technology, but aggregate as if the elasticity of substitution among factors in production is above one, triggered by an IISTC. In addition, my framework features endogenous entry and exit (while in Aghion et al.(2019)), the number of firms is fixed), which obtains a more realistic firm size distribution and life cycle dynamics of firms. Consequently, my model is able to generate rising concentration and declining firm entry simultaneously. Finally, this paper contributes to the literature on studying the macroeconomic implications of intangible investment. Corrado, Hulten, and Sichel(2005) propose a framework to estimate for the values of intangible investments, a substantial part of which are unmeasured by national accounts. They consider three broad categories: computerized information including software and customized database, R&D and intellectual property, and economic competencies including brand, firm-specific human capital9 and organizational structures. National accounts like BEA, measure software, artistic originals, and, most recently, R&D as investment only. Corrado et al.(2016) estimate the intangible investment as a share of gross value-added to be 0.15 for business sector in 2016, while for BEA-NIPA, this number is only 0.055. The measurement of intangible investment has important macroeconomic impli-

9Human capital is measured by the costs of developing workforce skills such as the on-the-job training and tuition payments for job-related education

7 Firm state Exit Exit Entry Rent physical capital Produce Payments Div.

0 Draw z Exog. Endog. Intangible: kI0 = 0 kT ≤ ϕa y(z, kT 1, kI , l1) RT (kT 1 + kT 2) a ≥ 0

(kI , a, z, zI ) by πd Net worth: a0 xI (z, kT 2, kI , l2) w(l1 + l2 + κe) kI ≥ 0 d ≥ 0 C,B0,S0

Figure 1: Timeline of the Model cations. McGrattan and Prescott(2010b) illustrate the macro measurement such as GDP and income shares depends on how intangibles are measured and allocated between labor income and capital income. Karabarbounis and Neiman(2018) conveys similar information by studying the implications of the residual payments – or what they label “factorless income” that arises because the imputed payments to capital do not rise sufficiently given a period to fully offset the measured decline in payments to labor. Allocating factorless income differently (i.e. to profits, unmeasured capital, or opportunity cost of capital) has different consequences for understanding various macroeconomic trends such as the economy’s production technology, the evolution of competition across firms, measured labor share, and measured TFP. I contribute to the literature by exploring the macro implications of measured intangible investment on not only measured income shares, but also concentration and firm dynamics, driven by an IISTC. More specifically, my model is able to capture the empirical fact that firms with higher intangible-investment to-total-assets ratio are larger on average. An IISTC favors more intangible-intensive firms that are large, making them become even larger.

3 Model

Time is discrete in infinite horizon. There is a continuum of firms that are perfectly competitive in the output goods market and produce both the consumption/physical investment goods and intangible investment goods. Firms own tangible assets, accumulate intangible capital through producing intangible investment goods and rent physical capital subject to a multiple of tangible assets. Firms are also subject to persistent shocks to individual productivity, which, together with endogenous entry and exit, yield substantial heterogeneity in production. Households are identical and infinitely-lived. I abstract from aggregate uncertainty and

8 consider a stationary industrial equilibrium. Before describing firms’problem in detail, I outline the precise timing of the model, summarized in Figure1. Within a period, the events unfold as follows: (i) realization of the productivity shocks for incumbent firms; (ii) endogenous and exogenous exit of incumbents; (iii) realization of initial productivity and entry decision of potential entrants; (iv) production and revenues from sales; (v) payment of wage bill, operation expenses, and physical capital rental costs; (vii) firm’s decisions on dividend payment, intangible capital and tangible assets for next period, and household consumption/saving decisions. In the following sub-sections, I present my model economy and examine the optimization problems of firms, followed by the household problem and the definition of stationary competitive equilibrium and aggregation.

3.1 Production Technology

Each individual ﬁrm produces consumption/physical investment goods y and intangible investment goods xI using two types of capital - physical and intangible capital, and labor according to the following technologies:

1−α η (1−µ) µ α y = Az kT 1 kI (l1) (1)

and

1−α η (1−µ) µ α xI = AI zI kT 2 kI (l2) (2) where z is a firm-specific productivity shock on producing consumption/physical investment goods, following a Markov process, and zI is a firm-specific productivity shock on producing intangible investment goods10, which is time-invariant11 and unevenly distributed across firms in the economy. A possible source of persistent differences in productivity on producing intangibles is their business processes. Consider Amazon versus Barnes & Noble. While

10 Note that there are two types of labor input, l1 and l2 used to produce diﬀerent types of goods. In this set-up, I implicitly assume a frictionless labor market in which labor is a homogeneous input and workers can transition freely between sectors for producing consumption/physical investment goods and for producing intangible investment goods. 11 As a robust check, making zI a persistent shock, following a Markov process as well, does not alter my results. I target the persistence of this Markov process to match the persistence of intangible-investment to total assets ratio at the ﬁrm-level from the data, which turns to be a number close to 1.

9 Barnes & Noble relies on its chains of physical stores, Amazon developed an online platform to sell books. The different business processes employed by the two companies determine the different amount of resources they devote to investing on intangibles. Moreover, firms like Amazon have established successful business models and logistics that are evidently hard to copy and reverse engineer.

AI is the aggregate productivity on producing intangible investment goods, which is the driving force for the increased intangible investment relative total income. I call an increase in

AI relative to the aggregate productivity on consumption/physical investment goods A as an intangible-investment-speciﬁc technical change (IISTC). η is assumed to be 0 < η < 1, which implies decreasing returns to scale at the ﬁrm level12.

Note that kI , as an input to both types of goods, is not split between them as is the case for physical capital and labor. An existing software is used both to sell/produce ﬁnal goods and services and to develop new softwares. Patents are used both by the producers and the researchers.

3.2 Financial Friction

Firms face two financial constraints. First, the capital decision involves borrowing capital from financial intermediaries (banks) in intra-period loans. Because of imperfect contractual enforcement frictions, firms can appropriate a fraction 1/λ of the capital received by banks, with λ > 1. To preempt this behavior, a firm renting kT units of capital is required to deposit kT /λ units of the collateral with the bank. This guarantees that, ex post, the firm does not have an incentive to abscond with the capital. I assume that only tangible assets can serve as collateral, and likewise, intangible capital cannot be liquidated if the firm exits because business lending against intangible assets is usually difficult13. This is is consistent with the the empirical evidence that only a very few intangible assets (patents and brands) can be used as collateral and only an extremely small minority of secured syndicated loans (about 3% of

12One could interpret this as the consequence of “span of control” with some limited entrepreneur’s management skill (see Lucas(1978)). As a result, there will be a non-trivial firm heterogeneity in the stationary equilibrium and the most productive firms will not take all of the market. This is important for our purpose of studying firm concentration. 13The main reason is that there are limited, and sometimes no markets on which intangible assets can be readily sold to other potential users. Intangible assets are either too firm-specific (e.g. human capital) or not easily separated from the firms and transferred to other users (e.g. proprietary databases or software, or in-process R&D).

10 14 total loan value ) do use them as collateral. Thus, firms face collateral constraints kT ≤ λa. Second, I assume that firms may only issue equity upon entry: an incumbent must keep nonnegative dividends payments. The model requires both constraints. Otherwise, the collateral borrowing constraint can be easily circumvented and plays no role. The non-negative dividend constraint captures two key facts about external equity documented in the corporate finance literature. First, firms face significant costs of issuing new equity15. Second, firms issue external equity very infrequently16. The specific form of the non-negativity constraint is widely used in the macro literature because it allows for efficient computation of the model in general equilibrium17. An alternative set-up is to introduce costly equity issuance18, which plays the same role as the non-negative dividend constraint19. It should be noted that financial friction is introduced because it matters for quantitative results. Adding such a financial friction contributes to a more realistic firm life cycle since it hinders the births of start-ups and slows the expansions of young firms. Otherwise, firms jump to their optimal scale almost immediately after their births. This implies that statistics related to firm dynamics such as entry and concentration generated from a model without financial frictions are going to be very sensitive to the change in the value of technology parameters. Since a main goal of this paper is to study the impact of a technical change that drives intangible investment on firm dynamics, having a model that features realistic firm distribution and life cycles are important20. As in Figure2, my model produces an average life cycle of firms that is very close to its empirical counterpart21.

14Based on the results from Falato et al.(2018) that uses a large sample of syndicated loans to US corpo- rations for which a detailed breakdown of types of collateral used is available. This is also in line with Wang (2017). 15Costs of issuing new equity include both direct flotation costs (see, for example, Smith(1977)) and indirect costs (for example, Asquith and Mullins(1986)). 16See, for example, DeAngelo, DeAngelo, and Stulz(2010) 17See Khan and Thomas(2013), Khan, Senga, and Thomas(2017), Ottonello and Winberry(2018), among others 18This includes proportional costs of equity issues (e.g., Begenau and Salomao(2019); Cooley and Quadrini (2001); Gomes(2001); Hennessy and Whited(2005)) and quadratic costs (e.g., Hennessy and Whited(2007)) 19As a robust check, I allow part of the firms (corresponds to public firms) to issue equity at a cost and find there’s almost no change in the quantitative results. See Table8 in subsection 5.2 20From this perspective, the assumption that only tangible assets can be used as collateral is not essential. It is made to be consistent with the empirical evidence. Making intangible capital as collateralizable as tangible assets will not alter the results on concentration and firm dynamics. I will show this result in section 5 21I admit that there are other set-ups that may also be able to achieve this goal. A typical example is to make physical capital accumulated rather than rented and introduce physical capital adjustment cost instead of the collateral constraint.

11 3.3 Entry and Exit

I model firm entry and exit based on the standard approaches in the literature22. In each period, incumbent firms may exit the economy either by an exogenous probability or by their endogenous decisions. Due to the financial frictions at the firm-level, individual states of productivity, intangible capital, and tangible assets jointly affect the latter endogenous exit decision. Together with endogenous entry decisions by potential entrants, my model features realistic firm dynamics.

Entry

At each period, there is a ﬁxed mass of potential entrants M0. Potential entrants draw a

productivity signal on consumption/physical investment goods z0 from the ergodic distribu-

tion Γ0 that will inform the productivity when they enter the market and begin to produce.

Similarly, they draw a productivity on intangible investment goods zI0 which will be their real productivity on producing intangibles after they enter the market. They start with zero

intangible capital and the same positive amount of tangible assets a0, which is ﬁnanced by households in terms of equity injection. This is the only time when ﬁrms are able to issue equity.

They decide whether to enter the market by paying the fixed entry cost κe, denominated in labor units, which can be interpreted as labor utilized for entry such as entrepreneurs in start-ups. In this way, I am able to set up a simple binary problem of endogenous entry. Note that firm entry takes place at the end of a period, and since they start with zero intangible capital, they need to invest on intangible capital in the first period they enter the market given their initial productivities, (z0, zI0), and start producing in the following period. Let

i (z0, zI0) ∈ {0, 1} denote the entry decision rule. The value of entry is:

0 0 Z U C e 0 0 0 0 0 v (z0, zI0) = max − kI + β 0 v kI , (1 + r) a0, z , zI dH z |z0 (3) 0 U (C) kI Z

22Hopenhayn(1992) is the seminal work in this literature with industry dynamics driven by ﬁrms’ endogenous entry and exit. Clementi and Palazzo(2016) modify the timing of entry in the Hopenhayn model to investigate the business cycle implications of ﬁrm dynamics. I follow the similar approach of them, while introducing the exogenous exit as employed in Khan and Thomas(2013).

12 where v0 (· ) is the value of of an operating ﬁrm at the beginning of each period, a function of

(kI , a, z, zI ). The ﬁrm chooses to enter the market if and only if the value of entry exceeds start-up 23 costs κe, denominated in labor units, plus household equity injection a0 , i.e.

e v (a0, z0) − wκe ≥ a0

Entrants start with relative low net worth compared to mature firms. Due to the fixed start-up cost denominated in labor units, an increase in wage rate (resulting from technological advances) may suppress firm entry24.

Exit

At the beginning of each period, ﬁrms are informed of their respective status of exit which

takes place before production. The ﬁxed probability of exit, πd, is common across ﬁrms. The

remaining firms after this exogenous exit need to pay κo units of labor in order to continue operation in the next period. This fixed cost of operation denominated in labor can be interpreted as overhead labor such as expenses on hiring HR and accountants, which creates a binary exit decision. If a firm does not pay this cost, it has to exit the economy permanently with liquidation value equal to its net worth in terms of tangible assets a. Thus, only firms continuing to the next period make inter-temporal decisions on investment and saving after

paying κo. Due to the fixed operation cost, denominated in labor units, an increase in wage rate will have larger impact on small firms. This is relevant for quantitative results, as I will show in section 5. It should be noted that both endogenous and exogenous exit are necessary elements to this model. With a fixed probability of exogenous exit, all firms have equal chances to exit,

23The treatment of the entry problem is different from that in Hopenhayn(1992). There, prospective entrants are identical. Selection upon entry is due to the fact that firms that paid the entry cost start operating only if their first productivity draw is greater than the exit threshold. In my framework, prospective entrants are heterogeneous. Some obtain a greater signal than others and therefore face better short-term prospects. As a consequence, they have greater intangible capital and are likely to hire more employees and rent more physical capital when they start to produce. Thus, the entry conditions require both fixed mass of potential entrants and free-entry condition where firms pay a fixed cost to enter, in contrast to Hopenhayn(1992) requiring only the latter. This set-up is in line with Clementi and Palazzo(2016), Jo and Senga(2019), and Gavazza, Mongey, and Violante(2018), among others. 24This is in line with Khan, Senga, and Thomas(2017) showing that in a GE set-up similar to my framework where heterogeneous firms face credit constraints, increased factor prices, due to credit subsidies, reduce the number of firms in production

13 regardless of firm size. This assumption helps the model reproduce the empirical distribution of firm size and firm age by allowing turnovers of large-mature firms (e.g. job destruction in large and mature firms). The endogenous exit margin of firm dynamics enables relatively less-profitable firms to endogenously choose to exit. Financially constrained and unproductive firms are more likely to exit and have higher job destruction rate. This is consistent with the empirical evidence that small and young firms have higher exit rate and job destruction rate in general. To validate this set-up, I check the job destruction rate25 of firm with different sizes in terms of employment as a non-targeted moment. Let v0 (· ) be the value of an operating firm at the beginning of the current period, before its survival from exogenous exit is known. Accordingly, define v1 (· ) as a surviving firm’s value, before making its decision to pay the operation cost κo in terms of labor. Finally, if a firm decides to continue to the next period, its value is given by v (· ) . Once a firm exits, its liquidation value equals its net worth in terms of tangible assets, a, due to the non- collateralizability of intangible capital. Then that firms can exit exogenously or endogenously can be summarized into the following problem:

0 1 v (kI , a, z, zI ) = πd· a + (1 − πd) v (kI , a, z, zI ) (4)

1 v (kI , a, z, zI ) = max {a, v (kI , a, z, zI )} (5)

Incumbent ﬁrm

The recursive form of the problem of incumbent ﬁrms is given by

0 0 U C 0 0 0 0 v (kI , a, z, zI ) = max d + βE 0 v kI , a , z , zI (6) 0 0 U (C) kI ,a ,l1,l2,kT 1,kT 2,d s.t.

0 d + pxI + a = y + pxI − wl − wκo − (r + δT )kT + (1 + r) a |{z} |{z} | {z } |{z} |{z} | {z } dividend intangible invest. NIP A income wage overhead labor rental cost

25I focus on job destruction rate rather than exit rate because very large ﬁrms usually "exit" due to merger that my model does not feature.

14 1−α η (1−µ) µ α y = Az kT 1 kI (l1)

1−α η (1−µ) µ α xI = AI zI kT 2 kI (l2)

0 kI = (1 − δI ) kI + xI , xI ≥ 0

kT = kT 1 + kT 2, l = l1 + l2

kT ≤ λa, d ≥ 0

For each individual ﬁrm, there is a shadow price of intangible investment goods xI relative to consumption/physical investment goods y, which is determined by26:

1 1 p = / (7) AI zI Az

. To help understand the budget constraint and preface how I take the model to the data,

define firm debt by the identity b := kT − a, with the understanding that b < 0 denotes savings. Making this substitution reveals an alternative formulation of the model in which the 0 0 firm owns its physical capital rather than rent it and faces a constraint on leverage: b ≤ θkT

where λ = 1/ (1 − θ). With state vector (kI , kT , b, z, zI ), the ﬁrm faces the following budget constraint27:

d + xT + pxI = y + pxI − wl − wκo −rtbt+ bt+1 − bt |{z} |{z} |{z} | {z } |{z} |{z} | {z } dividend physical investment intangible investment NIP A income wage overhead labor 4Borrowing

3.4 Representative Households

Assume that there is a unit measure of identical risk-neutral households in the economy. In each period, households consumes, supplies labor inelastically, make bank deposits, and invests in ﬁrms’ shares: 26I call it shadow price is because intangible capital is accumulated within each individual ﬁrm and there is no market for it to be traded, but there is still a shadow price of intangibles relative to consumption/physical investment goods which is determined by the relative productivities of the two types of goods. 27 By making a timing assumption such that z for period t + 1 is realized when making decisions on kT t+1, the problem with three endogenous three variables can be reduced to the current one with only a and kI , following Moll(2014) and Midrigan and Xu(2014), among others

15 0 0 W (B,S) = maxT 0 ,M 0 ,C>0U (C) + βW B ,S (8) subject to 0 0 C + QB¯ + PS = wN¯ + (D + P ) S + B where B are bank deposits, S are shares of the mutual fund composed of all firms in the economy, and D are aggregate dividends per share. The household takes as given the price of bank deposits Q¯ , the share price P , and the price of the consumption good, which is the numeraire, so normalized to 1. From the first-order conditions for deposits and share holdings, I obtain Q¯ = β and P = β (P + D), which imply a time-invariant rate of return of r = β−1 −1 on both deposits and shares. The household is therefore indifferent over portfolios. Because of risk neutrality, the household is indifferent over the timing of consumption as well28.

3.5 Recursive Competitive Equilibrium

P P P P I consider a stationary industrial equilibrium of the model. Let KI , A, Z and ZI be the Borel sigma algebras over KI and A, Z, and ZI . The state space for an incumbent P firm is S = KI × A × Z × ZI , and denote the individual state (kI , a, z, zI ) by s. Let S be P the sigma algebra on the state space, with typical set S = KI × A × Z × ZI , and (S, S) be the corresponding measurable space. Denote with ϕ : P S −→ [0, ∞) the stationary measure of incumbent firms at the beginning of the period, following the draw of firm-specific

productivity, before the exogenous exit shock. Accordingly, denote ϕe as the mass of actual entrants. For simplicity, denote ϕp as the distribution of producing ﬁrms that survives from exogenous shocks and decide to continue, and denote ϕex as the distribution of exiting ﬁrms (including both exogenous and endogenous exit).

To simplify the exposition of the equilibrium, it is convenient to use s ≡ (kI , a, z, zI ) and

s0 ≡ (z0, zI0) as the argument for incumbents’ and entrants’ decision rules. A stationary recursive competitive equilibrium is a collection of ﬁrms’ decision rules n 0 0 o 0 1 e kI (s) , a (s) , d (s) , kT 1 (s) , kT 2 (s) , l1 (s) , l2 (s) , e (s) , i (s0) , value functions v , v , v, v ,

a measure of entrants ϕe, a distribution of ﬁrms ϕ, wage w, policy functions for households

28Without loss of generality, I assume risk neutrality of households for simplicity. In this case, I only need to clear one market - labor market. Otherwise, I need to clear both asset market and labor market. Using other forms of utility function (e.g. GHH, risk-averse households, perfectly elastic/inelastic labor supply) will not change my results. See section 5 for a robustness check

16 n 0 0 o C,B ,S with the associated value function W that solve the optimization problems and clear markets in the following conditions.

n 0 0 o 1. The decision rules kI (s) , a (s) , d (s) , kT 1 (s) , kT 2 (s) , l1 (s) , l2 (s) , e (s) , i (s0) solve 0 1 e the ﬁrm’s problems (3), (4), (5), and (6), v , v , v, v are the associated value func-

tions, and ϕe is the mass of entrants implied by

Z Z ϕe = M0 i (z0, zI0) dUdΓ0 (9) Z ZI

n 0 0 o 2. W solves (8) , and C,B ,S are the associated policy functions.

3. Labor markets clear:

Z Z p e N¯ = (l1 + l2 + κo) dϕ + κedϕ (10) S S

4. Goods markets clear (resource constraints hold):

Z Z Z ex Ch + KT − (1 − δT ) KT + M0 i (z0, zI0) dUdΓ0 − adϕ = Y (11) Z ZI S

R p R p where Y = S ydϕ and KT = S kT dϕ

Z p XI = xI dϕ (12) S

5. Shares markets clear (by Walras’ Law) at M = 1 with share price

Z Z Z e P = v (s) dϕ + M0 i (s0) v (s0) dUdΓ0 S Z ZI

and aggregate dividends

Z Z Z Z D = πd adϕ+(1 − πd) {[1 − e (s)] d (s) + e (s) a} dϕ−M0 a0i (z0, zI0) dUdΓ0 S S Z ZI

6. The distribution of ﬁrms, ϕ, is a ﬁxed point where its transition is consistent with the

17 policy functions and the law of motion for ϕ, which is given by

R ϕ (KI × A × Z × ZI ) = (1 − πd) [1 − e (s)] 1 0 1 0 Γ (Z, z) dϕ S {k (s)∈KI } {a (s)∈A} I (13) R R +M0 i (s0) 1 0 1 0 dUΓ (Z, z) dΓ0 Z ZI {kI (s0)∈KI } {a (s0)∈A}

4 Parametrization

I numerically solve the model by using non-linear methods, and ﬁnd a stationary equilibrium where individual decisions are consistent with market clearing prices29. In subsection 4.1,I discuss how I map from the model to the data. In subsection 4.2, I calibrate the model to be consistent with the observed data in the U.S. business sector. Once the model is calibrated, in subsection 4.3, I explore the main cross-sectional implications of the calibrated model, which are going to inform the aggregate results in section 5.

4.1 Variables of interest

Once an equilibrium for the model economy is computed, I can compute variables that are either moments used in parameterizing the model or economic statistics relevant for validation and counterfactuals. Key moments for parameterizing the model are the implied income shares and firm distribution. To be consistent with BEA’s definition post-2013-revision, own-account intangibles (including software, R&D, and artistic originals) are considered as final output. Hence, labor income share (post-2013-revision) in the model is defined as labor compensation divided by the the gross value-added of the domestic business sector:

¯ R p R e wN w S (l1 + l2 + κo) dϕ + S κedϕ SN = = R p R p (14) Y + PXI ydϕ + pxI dϕ where P is the aggregate price of intangible investment goods in terms of consumption/physical investment. Correspondingly, labor income share (pre-1999-revision) in the model is deﬁned as:

wN¯ SN = (15) pre Y

29See Appendix C for computational details.

18 when own-account intangibles are not treated as ﬁnal output. The relative aggregate price of intangible investment goods is deﬁned as expenditures on intangibles divided by quantity QI , which is:

R p pxI dϕ P = (16) QI

where p is the shadow price of intangible investment goods xI relative to consumption/physical investment goods y for each individual firm, defined in equation (7). For aggregate measured income shares, I focus on the corporate sector due to two reasons: first, clearer measurement of labor income and profit30 Second, the way I model firms is more consistent with (non-financial) corporates. For firm distribution, I focus on all the employer firms in U.S. business sector where the corporate firms are just a subset. The way I deal with this discrepancy is as follows. Since corporates are usually very large firms31, I filter corporate firms based on firm size in terms

of ﬁnal output and choose a cutoﬀ for y + pxI , call it v¯, such that the total income generated

by all the ﬁrms with y + pxI ≥ v¯ divided by the total income generated by all the ﬁrms in

the economy (i.e. Y + PXI in equation (14)) equals the corporate income as a share of the domestic business income from BEA-NIPA32. Any moments generated from my model that are used to calibrate parameters to match their empirical counterpart from Compustat also use this filtering criterion. I compare the national account generated from the calibrated model with the BEA-NIPA. The two are close to each other (See Table9). The starting point of the time-series considered in the paper is 1975 to ensure better estimates and consistency for intangible investment because 1975 is the first year that the Federal Accounting Standards Board (FASB) requires firms to report R&D.

30For corporate sector, labor income is unambiguously defined as the compensation of employees, but for noncorporate sector such as partnership or sole proprietorship, since owners also use their own time to contribute to the production, part of the proprietor’s income should be allocated to labor income. The exact magnitude of the decline in labor income income is affected by the treatment of the labor portion of proprietor’s income (See Gollin(2002), Elsby, Hobijn, and Sahin(2013)). 31For example, based on Dyrda and Pugsley(2019) that uses U.S. census data, 20% of all firms are corpo- rations, accounting for 90% of all sales in early 1980s. 32It should be admitted that there exists discrepancy between the coverage of BEA-business sector, which covers both employer and non-employer firms and the coverage of BDS/LBD, which only covers employer firms. However, since non-employer firms do not contribute to the total employment and only takes a very small portion in terms of sales (which is 2.48% based on SBO 2007), I assume they cover the same firms. See Table 12 for a comparison among the datasets for the coverage of firms.

19 Parameter Description Value Source/Target β Discount factor 0.990 Annual risk-free rate = 4%

M0 Mass of potential entrants 0.002 Measure of incumbents = 1 δT Physical capital depreciation rate 0.050 NIPA ﬁxed asset tables δI Intangible capital depreciation rate 0.215 NIPA ﬁxed asset tables

Table 1: Parameter values set externally

4.2 Calibration

I begin with the subset of parameters calibrated externally, and then consider those estimated within the model. Data moments are averages over 1980–1985 unless otherwise speciﬁed33.

4.2.1 Externally Calibrated

The model period is one quarter. I set β to replicate an annualized risk-free rate of 4 percent.

Since the measure of potential entrants M0 scales the distribution of entrants ϕe (see equation

(10)), I choose M0 to normalize the total measure of incumbent firms to 1. I use BEA fixed asset tables34 which include both flows and stocks to compute the depreciation rate of physical

capital δT = 0.05 as well as the depreciation rate of intangible capital δI = 0.215. Table 1 summarizes these parameter values.

4.2.2 Internally Calibrated

Table 2 lists the remaining 17 parameters of the model that are set by minimizing the distance between an equal number of empirical moments and their equilibrium counterparts in the model35. It also lists the targeted moments, their empirical values, and their simulated values from the model. Even though every targeted moment is determined simultaneously by all parameters, in what follows I discuss each of them in relation to the parameter for which, intuitively, that moment yields the most identiﬁcation power. There are two key sets of parameters to be calibrated within the model: those that characterize the production technology and those that characterize the heterogeneity across ﬁrms

33Ideally I would only use data on firms, since financial constraints apply at the firm level. However, since some moments are only available at the establishment-level and my model indeed does not differentiate between firms and establishments, I use firm data whenever I have a choice, and establishment data only when firm-level data is not available. 34For more details, see Appendix A 35Specifically, the vector of parameters Ψ is chosen to minimize the minimum-distance-estimator criterion 0 function f(Ψ) = (mdata −mmodel(Ψ)) W (mdata−mmodel(Ψ)), where mdata, mmodel are the vectors of moments 2 in the data and model, and W = diag(1/mdata) is a diagonal weighting matrix.

20 Parameter Value Moment Data Model Production Technology - consumption/physical investment goods & intangible investment goods Labor share α 0.670 Labor compensation/value-added 0.640 0.640 Intangible capital share µ 0.225 Intangible investment/value-added 0.032 0.032 L H Permanent Productivity on intangible invest. goods zI ∈ zI , zI H L High - low gap zI /zI 3.990 Share of sales going to top 10% H 0.519 0.493 Mass: zI ﬁrms µ H 0.050 Intangible-intensive ﬁrms zI Firm Size Distribution

Low DRS in prod. ηL 0.750

Mid DRS in prod. ηM 0.800

High DRS in prod. ηH 0.905Firm size distribution (BDS) see Table 3

Mass: ηL ﬁrms µL 0.690

Mass: ηH ﬁrms µH 0.065 Persistent Productivity on Consumption/Physical Invest. Goods z (AR1) Autocorrelation coeﬃcient 0.170 0.170 Persistence ρz 0.795 for log revenues

Standard deviation σz 0.296 Std. of log revenues 0.540 0.579 Entrants Start-up debt/value-added rel. to Initial tangible asset a0 1.210 1.739 1.645 aggregate debt/value-added Average start-up size rel. to Initial productivity (mean) zˆ0 0.391 3.384 3.387 average incumbent size Financial Friction Collateral parameter-low value λ 8.500 Aggregate debt/value-added 0.608 0.605 Entry and Exit

Exog. exit prob. πd 0.075 5-year survival rate 0.485 0.547

Entry cost ξe 6.000 Annual entry rate 0.125 0.114

Operating cost ξo 0.035 Average ﬁrm size 20.47 20.66

Table 2: Parameter calibrated internally

Population Share (%) Employment Share (%) Employees Data Model Data Model 1 to 19 88.97 88.97 21.35 22.61 20 to 99 9.30 9.30 18.19 22.06 100 to 499 1.41 1.41 13.45 18.22 500 to 2499 0.24 0.24 11.27 11.17 2500+ 0.08 0.08 35.74 25.94

Table 3: Firm Size Distribution: Model v.s. Data (p.p.)

21 in the corresponding productivity state variables. I start with disciplining the parameters in the production technology for both consumption/physical investment goods (1) and intangible investment goods (2)36. I assume both technologies take a Cobb-Douglas structure and have the exactly the same parameters of factor shares because I don’t have additional information on whether, and the degree to which, these inputs are substitutes or complements, following the literature involving intangible capital as a production input37. The labor share α and the intangible capital share µ are chosen so that the model predictions for labor compensation as a share of gross value-added and the intangible investment as a share of gross value-added38 are consistent with the BEA data. The other key set of parameters is calibrated to capture the empirical fact that a small portion of firms account for a large share of total final output or employment due to two major reasons. First, a major goal of this paper is to study the impact of technological shocks on concentration, so having firm distribution matched to the data at the initial steady state is important. Second, how well this model is able to match the firm distribution determines how well this model is able to capture a key relationship (i.e. a negative correlation between firm size and firm-level labor share documented by Autor et al.(2017)) that will affect the aggregate results on concentration and labor share, as I will show later. In particular, I do two things to match firm distribution. First, I calibrate the distribution of productivity on producing intangible investment goods zI to capture the fact that intangible-intensive firms are large on average. In the baseline case, I consider zI follows a two-point distribution with support L H zI , zI to target the moment that top 10% firms in terms of intangible-investment-to-total- assets ratio accounting for 51.9% of total sales in early 1980s39. In subsection 5.2, I allow

36 For simplicity, I refer to the revenue function as if it were a production function: η represents the span of control. That is, I do not take a stand on whether z represents demand or productivity shocks, or whether η < 1 is due to DRS in production or downward-sloping demand. Given the model set-up, the revenue function is sufficient since it would not be the case in alternative environments that endogenize components of revenue productivity (e.g. Atkeson and Burstein(2010), Acemoglu et al.(2018)). 37See Bhandari and McGrattan(2019); McGrattan and Prescott(2010a,b), among others. 38Following the assumption from Karabarbounis and Neiman(2014) that the ratio of the nominal value of the capital stock to nominal investment is constant and that the required rate of return on capital is constant (See Section IV.B of Karabarbounis and Neiman(2014)), together with the depreciation rate of intangible capital computed using BEA data consistently, I can measure the intangible capital share as the ratio of intangible investment to gross value-added. 39This is a moment I document from Compustat. As I’ve mentioned before, I’m trying to target the whole demographics of firms in the U.S. economy-not only the very large public firms, to fix this discrepancy, I filter Compustat firms based on the same criterion as the corporate firms. That is, after I do the simulation, I choose a cutoff for y + pxI , call it v¯, such that the total income generated by all the firms with y + pxI ≥ v¯ divided by the total income generated by all the firms in the economy equals the corporate income as a share of the domestic business income from BEA-NIPA, and then I compute the relevant moments only for firms

22 zI taking more values, and compare the results with the baseline case. Second, I introduce permanent heterogeneity in the scale parameter η to match the skewed ﬁrm size distribution in terms of employment from the Business Dynamic Statistics (BDS). I consider a three-point

distribution with support {ηL, ηM , ηH }, leaving ﬁve unknown parameters: (i-iii) the values of

ηL, ηM , ηH ; and (iv)–(v) the fractions of low and high DRS firms µL, µH . This heterogeneity dramatically improves the results on matching the skewed firm size distribution40, see Table 41 4 . As I will show in subsection 5.2, without either heterogeneity in zI or heterogeneity in η such that firm size distribution cannot be well matched, the aggregate results on labor share and concentration will be significantly affected through a much more weakened negative relationship between firm size and firm-level labor share. The persistent firm-specific productivity of producing consumption/physical investment 0 0 0 2 goods z follows an AR(1) process in logs: logz = logZ + ρzlogz + ε , with ε ∼ (−σz /2, σz).I choose the persistence of the productivity process ρz = 0.795 so that the model generates an autocorrelation coefficient for log revenues equal to 0.8242. Similarly, I choose the standard deviation of productivity shocks σz = 0.296 so that the model generates a standard deviation of log revenues equal to 1.56, which is the average standard deviation in my data across industries and time. The initial productivity distribution on producing consumption/physical investment goods for entrants Γ0 is assumed to be exponential. The mean zˆ0 is chosen to match the average size of start-ups relative to that of incumbent firms. For the initial tangible asset a potential entrant can get from households a¯0, I calibrate it to match the start-up debt-to-output ratio relative to the aggregate debt-to-output ratio, which is 1.447/0.832 = 1.73943. Calibrating with y + pxI ≥ v¯. For measurement of intangible capital at the firm-level, see Appendix A.2 40This method follows Gavazza, Mongey, and Violante(2018). Permanent heterogeneity in productivity might also be used to match these facts (see Elsby and Michaels(2013); Kaas and Kircher(2015), but heterogeneity in η also generates small old firms alongside young large firms, thus decoupling age and size, which tend to be too strongly correlated in standard firm dynamics models with mean reverting productivity as Gavazza, Mongey, and Violante(2018) sugguest. An alternative way to match the firm-size distribution is to employ a non-Gaussian process for the persistent firm-specific productivity shocks (see Jo and Senga(2019)). 41Since employment is not a state variable in my model, there is no way to know the number of employees of each firm directly. I divided firms into five groups based on the population share from the data (this is why the model can perfectly match the data in terms of the population share) and then I can get the employment cutoff for each group of firms. Finally, I can compute the employment share of each group of firms from my model and compare the results with the data. 42The value of the persistence in this paper falls into the range of the persistence estimates (0.757, 0.966) in Foster, Haltiwanger, and Syverson(2008). 43 The reason why for a¯0, I target the relative ratio rather than the direct ratio is because for start-up debt to output data, I only have it for the year 2004, which is from the Kauffman Survey (See Robb and Robinson (2014)). Assuming the relative relationship between start-up debt to output and aggregate debt to output is

23 Moment Data Model Source Employment share: Age ≤ 1 0.036 0.042 Employment share: Age ∈ (1, 10] 0.297 0.201 BDS Employment share: Age ≥ 11 0.667 0.757 Exit rate < 20/exit rate (> 500) 32.11 22.67 Regression coefficient of payroll-to-sales on firm size (sales) -0.664 -0.403 ADKPV Top 10% concentration (sales) 0.675 0.714 Compensation share in firms’ profits 0.856 0.789 BEA-NIPA Persistence of intangible capital/total assets 0.780 0.751 Compustat ADKPV: Autor, Dorn, Katz, Patterson, and Reenen(2017), LBD census data -Various sectors; regression coefficient within the range [−2.37, 0], with value-added weighted average -0.664

Table 4: Non-Targeted Moments

the initial conditions of entrants to match the data is important since it matters for firm distribution and the results on concentration and firm dynamics. Having start-ups too large compared to the data results in increased firm entry and declined concentration when the same technology shocks that drives intangible investment is fed into the model. For financial frictions, I calibrate the collateral parameter λ which is economy-wide to match the aggregate debt-to-value-added ratio for the private business sector. For the re-

maining three parameters regarding ﬁrm dynamics: exogenous exit probability πd, entry cost

e, and the operating cost κo, I calibrate them to match three moments from the data: 5-year survival rate, annual entry rate, and the average ﬁrm size in terms of employment.

4.3 Cross-Sectional Implications

I now explore the main cross-sectional implications of the calibrated model, at its steady-state equilibrium. Table 4 reports some empirical moments not targeted in the calibration and their model-generated counterparts. The model can replicate fairly well the distribution of employment by firm age, which was not explicitly targeted. The average exit rate of firms with less than 20 employees relative to that of firms with more than 500 employees generated from the model is not very far away from its empirical counterpart. This speaks to the importance of introducing exogenous death shock, without which the average exit rate of small firms stable overtime, by using the data for aggregate debt to output in 2004 and the average of 1980-1985, I can infer the start-up debt to output averaged for years 1980-1985.

24 relative to large will be a huge number close to infinity44. Most importantly, I show that my model can well replicate the negative relationship between firm size and firm-level labor share, documented by Autor et al.(2017), which is at the heart of generating the main results of this paper. More specifically, Autor et al.(2017) regress the payroll-to-sales ratio on each firm’s sales as a fraction of total sales using LBD Census data for six main sectors45 to get a range of the estimates on the regression coefficients [-2.37, 0]. I calculate a value-added-weighted average of this coefficient to be -0.664. Using the simulated data from the calibrated model46, I find a regression coefficient of −0.403, which is within the range and close to the weighted average. There are three key elements that drives the negative correlation between firm size and firm-level labor share: (1) heterogeneity in zI ; (2) heterogeneity in η; and (3) overhead labor. As I have mentioned before, the first two elements determines how well this model is able to match the skewed firm distribution, which determines how well it is able to replicate this key relationship that will affect the aggregate results on concentration and labor share (see subsection 5.2). Compared to the first two elements, overhead labor plays a relative minor role, in contrast to papers relying on the overhead cost set-up solely to generate this relationship47. The share of sales going to the largest 10% of firms is also a non-targeted moment but is well captured by the model. Moreover, the fact that labor compensation as a share of the sum of profit and labor compensation is close to its empirical counterpart confirms that the value of intangible share parameter ζ = µ (1 − α) η in producing intangible investment goods xI is chosen neither too high nor too low. This implies that the symmetric assumption that two types of goods y and

xI have the same technology parameters is reasonable. Consider an alternative set-up where

y and xI have different technology parameters and assume an extreme case where ζ = 0 which means firms cannot use their current intangible capital to produce xI , this number will be much higher since profits will decrease dramatically.

44There are also technical reasons for introducing the exogenous death shock. First, it is a simple way to avoid the situation where financial frictions are irrelevant when firms survive long enough (see Khan and Thomas(2013)). Second, without the exogenous death shock or if its value is very small, it is much harder to find the stationary distribution of firms. 45They are wholesale trade, finance, manufacturing, retail trade, utilities + transportation, and services. 46Here’s how I map from the model to the data: since the model abstracts from intermediate goods, there is no difference between sales and value-added. Moreover, since the firm-level census data still treats intangibles such as software and R&D as expenses rather than investments, I let y to be sales in my model. That is, I regress w (l1 + l2 + κo) /y on y/Y . I also check what will happen if I regress w (l1 + l2 + κo) / (y + pxI ) on (y + pxI ) / (Y + PXI ) instead. 47See, for example, Aghion et al.(2019); Autor et al.(2017); Hopenhayn, Neira, and Singhania(2019)

25 Figure 2: Average Life Cycle of Firms: Model v.s. Data

In addition, that the persistence of the intangible capital to total assets in the model is close to its empirical counterpart confirms that introducing idiosyncratic productivity on intangible investment goods as a permanent shock is a good idea48. Otherwise, the value of persistence generated from the model will be even smaller. In Figure2, I plot the average firm size in terms of employment over the life cycle generated from the baseline economy and compare it with the data. The model features a realistic life cycle dynamics due to two elements - financial friction and costly accumulation of intangible capital. The first element plays a dominant role. Without financial friction (when λ → ∞), firms jump to their optimal level almost immediately which will affect the aggregate results on concentration non-trivially in response to a technology shock. In Figure3, I plot the average firm size in terms of employment and value-added, growth rates of firms in terms of tangible assets, and intangible capital-to-physical capital ratio for firms with different productivity on producing intangible investment goods from birth to

maturity. Panel A, B, and C show that firms with high zI , those with (permanent) high productivity on producing intangibles, account for the upper tail in the size and growth rate distributions. These firms, which are sometimes called "superstar firms", take more advantage

48 See subsection 5.2 where I allow zI following AR(1) process and the quantitative results on concentration and labor share driven by the technical change do not change.

26 Figure 3: Average Life Cycle of Firms: high zI firms v.s. low zI firms of an intangible-investment-specific technical change. Together with Panel D, these figures imply that firms with higher zI are more intangible-capital intensive and larger (in terms of employment and value-added) as well.

5 Main Results

This section uses the calibrated model from section 4 to study the aggregate implications of an intangible-investment-speciﬁc technical change (IISTC). The main results are summarized in subsection 5.1 where I quantify the long-term impact of the IISTC on measured labor share and concentration. In subsection 5.2, I inspect the key mechanism.

5.1 The Aggregate Implications

The intangible-investment-speciﬁc technical change (IISTC) is modeled as a permanent increase in the aggregate productivity on producing intangible investment goods AIt to match the decline in the relative price of intangible investment goods from the data, which is constructed as the ratio of the price of investment in intangible capital including software, R&D, and artistic originals to the price of the bundle of consumption and investment in traditional

27 physical capital including structures and equipment49, as shown in Figure4. The relative aggregate price of intangible investment goods at the initial steady state,

Pinitial, is determined by equation (16), which is

R p pinitialxI,initialdϕ Pinitial = QI,initial

The relative aggregate price of intangible investment goods at the new steady state after the 50 technical change, Pnew, is given by

R p pnewxI,newdϕ Pnew = QI,new

The nominal expenditures on intangible investment in terms of consumption/physical investment is straightforward, but how to separate price P from quantity QI depends on how the

intangible investment goods production xIt by each individual ﬁrms are aggregated. If the

xIt’s are perfect substitutes, then they only need to be added up, but if they are imperfect substitutes, we need to employ a CES aggregator with an assumed value of elasticity of sub-

stitution. Here I assume the intangible investment goods xIt produced by each individual firm 51 R p are prefect substitutes so that QI = S xI dϕ . This is equivalent to a CES aggregator with ε ε−1 1−ε R ε p elasticity of substitution going to infinity, i.e. QI = S xI dϕ with ε → ∞. Note that the lower the value of ε, meaning xI ’s produced by individual firms are less substitutable, the less sensitive the aggregate relative price of intangible investment to the technology shock, which means the more I need to increase the aggregate productivity on producing intangible investment goods AIt in order to match the decline in the relative price of intangible investment from data. For example, with ε → ∞, I increase AIt by 70% to match the data, while with ε = 3, I need to increase AIt by around 78% to match the data. Thus, for this reason, the resulting numbers reported in Table5 and6 are conservative predictions for the impacts of the IISTC on factor shares of income and firm concentration. 49For more details about the construction of the relative price of intangible investment, see Appendix A. 50While BEA constructs price indices in chains, since I focus on steady-states-long-run effects, I choose to use the specific form for the relative aggregate price of intangible investment given in the text, rather than, R p pnewxI,newdϕ for example, Pnew = R p with fixed quantity but have the new price divided by the initial price. pinitialxI,newdϕ 51 I admit that the intangible investment goods xIt produced by each individual firm are more likely to be imperfect substitutes considering that a significant part of the intangibles in the data are very firm-specific such as customized software and in-process R&D. However, since I don’t have sufficient information to estimate the elasticity of substitution among them, I’d rather turn to the perfect substitutability assumption.

28 Figure 4: The Relative Price of Intangible Investment Goods (Data Source: BEA)

Start of sample End of sample Change (pp.) Model Data Model Data Model Data Labor share wL Y +PXI 0.640 0.640 0.613 0.594 -2.7 -4.6 (post-2013 revision) Labor share wL Y 0.671 0.665 0.660 0.641 -1.1 -2.4 (pre-1999 revision)

Table 5: Aggregate Implications: Measured Labor Share

Start of sample End of sample Change (pp.) Category Model Data Model Data Model Data Annual Firm Entry rate 11.0 12.5 8.1 8.0 -2.9 -4.5 Employment Share: 37.1 47.0 41.3 51.7 +4.2 +4.7 Large Firms (500+) Employment Share: 75.7 66.7 82.6 80.5 +6.9 +13.7 Mature Firms (11 years +) Top 10% concentration (Sales) 71.4 67.5 75.8 72.8 +4.4 +5.3

Table 6: Aggregate Implications: Concentration

29 Initial steady state New steady state

High zI firm sales share 0.776 0.839 High zI firm intan. invest share 0.950 0.968 High zI firm value-added share 0.786 0.854

Table 7: Expansion of Firms with High zI

I report the main results in Table5 and6. The start of the sample I am targeting is early 1980s and the end of the sample is 2010s. For measured labor share, the data I am targeting is the year 1980 (initial steady state) versus the year 2016 (new steady state with the technical change) in the linear trend of the BEA-measured labor share for years 1975-201652. As shown in Table5, the intangible-investment-specific technical change (IISTC), or fallen relative prices of intangible investment good is able to explain both the decline in BEA-measured labor share of post-2013 revision and of pre-1999 revision when iinvestments into intangible capital were treated as expenditures rather than investments, by around 50%. This is consistent with the increased intangible investment as a share of gross value-added, which implies that the drop in the aggregate labor income share would be smaller if intangibles are not treated as final output. For statistics related to concentration, the data I’m focusing on is the average of 1980- 1985 (initial steady state) versus the average of 2011-2016 (new steady state with the technical change). For concentration, I look at three measures: (1) the employment share of large firms with more than 500 employees; (2) the employment share of old firms with more than 11 years; and (3) the share of sales going to the largest 10% of firms53. As shown in Table6, the IISTC, targeted to match the decline in the relative price of intangible investment goods, accounts for around 2/3 of the decline in annual firm entry rate, almost 90% of the increase in the employment share of large firms, half of the increase in the employment share of mature firms, and about 80% of the rise of the market concentration.

30 5.2 Inspecting the Key Mechanism

To inspect the key mechanism, I do three things. First, I check how firm distribution changes due to the IISTC. Second, I explain why firm composition changes in a way that leads to the rise of concentration and the decline in labor share by deriving the aggregate elasticity of substitution numerically. Finally, I consider a number of extensions to the baseline model by altering its key features one by one to understand what are the most essential elements of the baseline model and what are the deep parameters driving the key relationship that generate the results in subsection 5.1. As shown in Table7, compared to the initial steady state, the share of firms with high productivity on producing intangible investment goods zI in terms of total sales, intangible investment, and value-added all increase. This confirms the intuition for increased concentration led by the IISTC: having high productivities on producing intangibles makes a firm large at the initial steady state, but the technical change makes those firms with high zI become even larger in the new steady state. As shown in Figure5, we can see that the decline of the aggregate labor share is driven by a strong reallocation of value added to firms with low labor shares due to the IISTC. This is consistent with micro evidence from Kehrig and Vincent(2018) 54. Next, I derive the aggregate elasticity of substitution numerically. Since there is no information about what the aggregate production function looks like in my model, I compute the aggregate elasticity of substitution between factors in production as follows. First, I compute

the aggregates of factor input - physical capital stock KT , intangible capital stock KI , and

labor N as well as the factor prices RT ,RI , w for the three types of inputs respectively at both initial steady state and new steady state with the IISTC. Then the aggregate elasticity

52Choosing 1975 as the starting year is to ensure better estimates and consistency for intangible investment (because 1975 is the first year that the Federal Accounting Standards Board (FASB) requires firms to report R&D), and hence, the measured labor share. 53The data on the first two measures comes from BDS. The data on concentration in terms of sales comes from Autor et al.(2017) for various sectors weighted by industry sales shares, and the empirical moment is the average of the years 1987-1992. 54See Figure 1 in Kehrig and Vincent(2018)

31 Figure 5: Output Reallocated Towards Firms with Low Labor Share

of substitution between KT and KI is computed using a mid-point rule as follows:

KI − KI KT final KT initial K K K I 1 I + I dln K 2 K K σ = T ∼ T final T initial KIKT = R R RT mid point rule T − T dln RI RI RI final initial 1 RT + RT 2 RI final RI initial

The aggregate elasticity of substitution between intangible capital KI , and labor N , σKI ,N , and between physical capital KT , and labor N , σKT ,N are computed in the same manner. The results are

σKI ,KT = 1.383, σKI ,N = 1.288, σKT ,N = 1.282

. We can see that the aggregate elasticity of substitution between factors in production all exceeds one. That is, in my economy, although individual firms operate at the Cobb-Douglas technology, they aggregate as if the elasticity of substitution is more than Cobb-Douglas driven by the IISTC. This explains why the firm distribution shifts towards more intangible-intensive firms with low labor share due to the IISTC. Finally, to understand what are the most essential element of the baseline model and what are the deep parameters driving the key relationship that generate the results on concentration and labor share led by the IISTC, I consider a number of alternative set-ups that may or may

32 Cross-sectional Moment Macro Moment

Emp. share Market share Regression coefficient Cases large firms top 10% firms labor share on 4 Concentration 4 Labor share (500+) (intan. invest.) firm size emp. sales entry Data 47.0 51.9 -0.66 4.7 5.3 -4.5 -4.6 Baseline 37.1 49.3 -0.40 4.2 4.4 -2.9 -2.7 No heterog. η 12.3 42.4 -0.05 0.5 0.5 +0.1 -0.9

No heterog. zI 23.8 25.6 -0.29 1.2 1.3 -0.2 -1.5

zI ∼ AR(1) 37.2 50.1 -0.41 4.3 4.5 -2.9 -2.7 No overhead 36.2 49.1 -0.38 4.1 4.2 +0.0 -2.5 λ → ∞ 50.2 65.8 -0.44 26.9 22.2 +10.0 -2.8

kI collateral 37.2 49.5 -0.40 4.2 4.4 -2.9 -2.7 Equity issu. 36.9 49.1 -0.40 4.2 4.4 -2.9 -2.7 Risk-averse 36.8 49.4 -0.40 4.1 4.2 -2.9 -2.7 HHs

Table 8: Alternative Set-ups not change the results. The cases I am considering are: (1) no heterogeneity in η; (2) no heterogeneity in zI ; (3) persistent zI shocks (AR1); (4) no overhead labor; (5) no ﬁnancial

friction (λ → ∞); (6) making intangible capital kI as collateralizable as tangible assets; (7) allowing equity issuance at a cost; and (8) risk-averse households. I list the results in Table 8. I recalibrate parameters in all these experiments to reproduce the same set of moments in the data as I have done in the baseline economy except for the first two cases without which skewed firm size distribution cannot be matched well. As I have discussed in subsection 4.2, heterogeneity in the span of control parameter η and in productivity on producing intangible investment goods zI are introduced to match a set of empirical facts capturing that a small portion of firms account for a very large share of total employment and total final output. Without either, these empirical facts cannot be well matched, thus denting the negative correlation between labor share and firm size. The direct consequence is that the magnitudes of both the increase in concentration and the decline in labor share are much smaller compared to the baseline.

In the baseline economy, productivity on producing intangible investment goods zI is a permanent shock. Modeling it as a persistent shock following an AR(1) process and intro-

33 ducing more grids55 do improve the results but to a very limited extent. Overhead labor contributes to better results as well, to a limited extent on concentration in terms of sales and employment, but improve the results on firm entry significantly. The next three cases are about financial frictions. In the first one, I let the collateral parameter λ go to infinity so that there is no financial friction. In that case, the moment on employment share of large firms and market share of top 10% in terms of intangible- investment-to-total-assets ratio are overshoot. Since this set-up does not feature a realistic life-cycle of firms in the sense that firms jump to their optimal level almost immediately after their birth, the moments on concentration are very sensitive to the value of technology parameters. An improved aggregate technology on producing intangibles disciplined by the decline in the relative price of intangible investment goods increase the concentration in terms of sales and employment drastically. The increase in the model by percentage point is almost six times as large as in the data. Moreover, it makes the annual entry rate increase by 10 percentage points, while its empirical counterpart declines by 4.5 percentage points. These results imply that having a model with realistic life cycle of firms is important to study 56 concentration and firm dynamics . In the second case when intangible capital kI is made to be equally collateralizable as tangible assets so that the collateral constraint becomes kT ≤ 57 λa + (λ − 1) kI . Such financial friction also contributes to a realistic life cycle of firms so there is almost no change on the results of concentration compared to the baseline case. The implies the non-collateralizability assumption of intangible capital is not crucial to explain the change in firm concentration (or labor share)58. Since in my mode, intangible capital is accumulated within firm and cannot be traded, that intangible capital has no collateral value seems to be a more natural assumption. I choose my baseline set-up also to be consistent with the empirical evidence as well as the existing literature (see Ai et al.(2019); Caggese and

55 I increase the number of grids of zI from two (in the baseline) to five. 56In Aghion et al.(2019), the concentration in terms of sales share of top 10% firms increases by 35.1 percentage points (while the data is 5.3 p.p.) driven by fallen firm-level costs of spanning multiple markets matched to the change in between component of labor share from the data 57 0 0 0 This is derived from: b ≤ θ kT + kI where b := kT − a, plus the timing assumption following Moll (2014) and Midrigan and Xu(2014). 58However, if the focus of the paper is to explain the debt-financing patters of firms with different intangible intensity, firms’ cash holdings, or the cyclicality of equity returns, this assumption will be the key to the results. See, for example, Wang(2017), Falato et al.(2018), and Ai et al.(2019). Consistent with these papers, the baseline model featuring the assumption that intangible capital cannot be used as collateral contributes to explaining the rise of corporate saving flows driven by the intangible-investment specific technical change since early 1990s, but this is not going to be the focus on of my paper. Rise of corporate saving is a byproduct of the decline in labor share. See Chen, Karabarbounis, and Neiman(2017).

34 Perez-Orive(2018); Chen(2014); Falato et al.(2018); Garcia-Macia(2017); Wang(2017)). In the third case, I allow part of the firms (corresponding to public firms) to issue equity at a cost: d + H (d) 1{d≤0}where H (d) = −ζ|d| and ζ is chosen to match the equity-to-asset ratio in Compustat, while the rest of the firms are still facing non-negative payment condition. There’s almost no change in the quantitative results. In the last case, I consider risk-averse households where they face log utility function in consumption. We can see that this assumption does not affect the aggregate results on concentration and labor share. Thus, considering that assuming risk-averse households increases the computation burden non-trivially since I need to clear one more market (i.e. asset market) besides labor market, I decide to keep the risk-neutral households set-up as in the baseline economy. To sum up, by considering alternative set-ups, I verify three key features of the baseline model that contributes to the results in section 5: first, introducing heterogeneity in η is important; second, whether zI is a permanent shock or a persistent shock is not crucial but

heterogeneity in zI is important; Third, having ﬁnancial friction is necessary but whether enabling intangible capital collateralizable or not is not important. I choose it to be non- collateralizable to be consistent with the empirical evidence from the existing literature.

6 Conclusion

In this paper, I propose a GE framework of firm dynamics that can potentially account for three major trends of the U.S business sector over the past three decades: (i) increased intangible investment relative to total business income; (ii) declined measured labor income share; and (iii) increased concentration in large firms in terms of employment and sales at the national level. I show that a significant part of these phenomena can be explained by a rapid fall in the relative price of intangible investment goods, such as software, manifested as an increase in the aggregate productivity on producing intangible investment goods relative to that on consumption/physical investment goods, where I call it an "intangible-investment- specific technical change (IISTC)". In my theory, individual firms operate at Cobb-Douglas technology to produce both types of goods at heterogeneous productivity level, but they aggregate as if the elasticity of substitution in production is greater than one, triggered by the IISTC. Cross-sectionally, my model can well replicate the negative correlation between

35 firm-level labor share and firm size from Autor et al.(2017). Firms with higher productivity on producing intangible-investment goods are more intangible-capital intensive and larger and they benefit disproportionately from the IISTC. Consequently, firm distribution shifts towards large firms with low share, which leads to rising concentration and declining labor share simultaneously. By reconciling the empirical facts regarding labor share and concentration using both aggregate and census data, my work emphasizes the importance of a comprehensive approach that links changes in micro-level heterogeneity to macro-level outcomes in analyzing the drivers of prominent empirical trends in the U.S. business sector. Taken together, my results have two major broad implications. First, they highlight the role of intangible investment played on accounting for the trends of measured labor share. The declined measure labor share can reflect both technological change and improved measurement. The lack of attention to measurement can severely misguide economic theory. Because intangible investment is largely unobservable, its measurement is challenging (Bhandari and McGrattan(2019); Corrado et al.(2016); Corrado, Hulten, and Sichel(2005); Karabarbounis and Neiman(2018); McGrattan and Prescott(2010a,b)). However, since an essential aspect of the US macroeconomic model is the factor distribution of income which relies explicitly on the measurement of intangible investment, future research efforts should be devoted to reasonably defining the boundaries of intangibles, accurately measuring intangibles at both firm-level and aggregate-level as well as figuring out the factor distribution of intangible capital rents. Moreover, my results also indicate that a large fraction of the slowdown of the business dynamism (e.g. decline in firm entry) and rising concentration is the result of new technologies such as falling software prices that favor the intangible-intensive, highly productive firms who are more adaptable to transitioning towards a more intangible-intensive economy, thereby increasing their efficiency and advancing their market share and employment share dispro- portionally. Viewed through the lens of my model, increased concentration, declined firm creation, and overall dynamism of the U.S. economy should not be cause for concern. This is complementary to the recent heated debate about the potential sources of rising concentration and declining business dynamism: rising mark-ups due to a diminution of competition, or the expansion of highly productive "superstar firms". This paper is in line with the second possibility.

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42 Appendix A: Data and Measurement

In this appendix, I describe the data used in constructing the empirical moments to discipline the model and the intangible-investment specific technical change (IISTC). Subsection A.1 provides the details of the sources and construction of the aggregate data series. In particular, I discuss how I construct the relative price of intangible investment goods in terms of consumption/physical investment goods in Subsection A.1.2. I also show that my model can capture the national accounts table well in Subsection A.1.3. Subsection A.2 is about the firm-level data I use. In particular, I discuss how I measure intangible capital at the firm-level to be consistent with BEA in subsection A.2.1.

A.1 The Construction of Aggregate Data Series

All the aggregate series are retrieved for the period 1975-201659. There are three sources of data that I use:

National Income and Product Accounts (NIPA-BEA) NIPA 1.7.5, NIPA 1.12, NIPA 1.13, NIPA 1.14, NIPA 2.3.3, NIPA 2.3.5, NIPA 5.3.4, NIPA 5.3.5

Fixed Assets Accounts (FAT-BEA) FAT 1.1, FAT 1.3, FAT 2.1, FAT 2.4, FAT 4.7

Flow of Funds Table L.102: Aggregate balance sheet data for the U.S

A.1.1 Depreciate Rate by Type of Capital

I construct the net stock of capital and depreciation of capital for traditional physical capital and for intangible capital (corresponding to IPP capital in BEA). Since I focus on the private sector only, the net stock of traditional physical capital is the private sector nonresidential structures, equipment, and residential capital. The net stock of IPP capital is only in nonresidential. The net stock of capital by type of capital (BEA-FAT 1.1, 2.1): Private IPP: KIPP 59Choosing 1975 as the starting year is to ensure better estimates and consistency for intangible investment (because 1975 is the ﬁrst year that the Federal Accounting Standards Board (FASB) requires ﬁrms to report R&D), and hence, the measured factor shares of income

43 Private physical: KT = KP,ST,NRes + KP,EQ,NRes + KP,Res The depreciation by type of capital (BEA-FAT 1.3, 2.4): Private IPP: DEP IPP Private physical: DEP T = DEP P,ST,NRes + DEP P,EQ,NRes + DEP P,Res The capital depreciation rate by type of capital is then:

DEP IPP δ = I KIPP and DEP T δ = T KT

A.1.2 Relative Price of Intangible Investment

I construct the relative price of investment in traditional physical capital and in intangible capital (corresponding to IPP capital in BEA). The price of the bundle of consumption/physical investment good is the numeraire. C C I ﬁrst construct the price index for consumption Pt . Let Pt be the price index for nondurable goods and service good i in year t, computed as the ratio between nominal i C i QC P C = Cit consumption of good , it, and the quantity index of good , it, i.e. t QCit , for i ∈ {ND,SV } sCi = Cit i . Let t CNDt+CSV t be the corresponding nominal share of good in period t. All the variables are from NIPA 2.3.3 and 2.3.5. Denote the growth rate of a variable xt λ (x ) = xt − 1 ≈ ln xt T ornqvist¨ to be t xt−1 xt−1 . Then, the growth rate of the price index for consumption is C C X s i + s i λ P C = t t−1 λ P Ci t 2 t i The level of the consumption price index is recovered recursively:

C C C Pt = Pt−1 1 + λ Pt

C where P0 is normalized to 1 at the initial period. Second, I construct the price of investment in traditional physical capital including struc- ST tures and equipment. For price of investment in structures Pt , I use price index for consump- C tion Pt constructed in step 1 as a proxy. In computing the price index of equipment invest-

44 EQ,Res ment, I use a T ornqvist¨ price index for private residential equipment investment, Pt , and EQ,NRes EQ,Res private non-residential equipment investment, Pt , from NIPA Table 5.3.4. Let st EQ,NRes and st be the share of private residential and non-residential equipment investment of total equipment investment using data from NIPA Table 5.3.5. The the growth rate of the price index of equipment is

! ! sEQ,Res + sEQ,Res sEQ,NRes + sEQ,NRes λ P EQ = t t−1 λ P EQ,Res + t t−1 λ P EQ,NRes t 2 t 2 t

Then in computing the price index of traditional physical investment, I use a T ornqvist¨ price index again for structures and equipment constructed above. The growth rate of the price index of the traditional physical investment is given by:

! ! sEQ + sEQ sST + sST λ P T = t t−1 λ P EQ + t t−1 λ P ST t 2 t 2 t

EQ ST where st , st are the share of equipment and structures investment of total traditional physical investment using data from NIPA Table 5.3.5. Third, I construct the price of consumption/physical investment bundle using results in sC = CNDt+CSV t ﬁrst two steps. Let t CNDt+CSV t+investEQ+investST be the corresponding nominal share of consumption goods in the sum of consumption and traditional physical investment in period

twhere investEQ, investST are nominal investment in equipment capital and structures capital sT = investEQ+investST respectively from NIPA Table 5.3.5. Similarly, Let t CNDt+CSV t+investEQ+investST be the corresponding nominal share of traditional physical investment in the sum of consumption and traditional physical investment in period t. Using a T ornqvist¨ price index again for consumption and traditional physical investment constructed above. The growth rate of the price index of the consumption/physical investment bundle is given by:

! ! sC + sC sT + sT λ P C,T = t t−1 λ P C + t t−1 λ P T t 2 t 2 t

Then the level of the price indices of consumption/physical investment bundle is recovered recursively as C,T C,T h C,T i Pt = Pt−1 1 + λ Pt

45 Fourth, I construct the price of investment in IPP, which is only available for non-residential I investment. I use the price index for IPP Investment Pt available in NIPA Table 5.3.4. As in C constructing Pt , normalize the price index of the IPP investment to 1 at the initial period as well. Finally, the relative price of investment (using the consumption/physical investment as numeraire) is deﬁned as I Pt pt = C,T Pt

A.1.3 National Accounts

The national accounts for the model can be expressed mathematically in terms of shares of ¯ R p income and product, where total income is Yb + Ynb where Yb = PY Y + PI XI = pyydϕ + R p ¯ ¯ pI xI dϕ represents business income and Ynb denotes nonbusiness income. Ynb and government expenditure G¯ are included as exogenous source of income. Gross Value-added (GVA) of corporate sector is deﬁned by:

Z Z p p GV A = (1 − τ) QC − PMtMt + τQC = pyydϕ + pI xI dϕ +τQ | {z } | {z } |{z} y+pxI ≥v¯ y+pxI ≥v¯ After tax Gross Output Intermediates Net taxes on production | {z } F inal output

Labor income of corporate sector is deﬁned by:

Z Z p e WLC =w ˜ (l1 + l2 + κo) dϕ +w ˜ κedϕ S where w˜ = Pyw Net Operating Surplus (NOS) of corporate sector:

Z p NOS = [pyy + pI xI − w˜ (l1 + l2 + κo) − py (r + δT ) kT ] dϕ y+pxI ≥v¯

Identity:

depreciation net prod. taxes Zz }| { p z}|{ PY YC + PI XIC + τQC = WL + NOS + δT kT dϕ + τQC | {z } |{z} GV A labor income | {z } Capital income

46 Data Model Income Shares R p R p ˜ Business income ydϕ + pxI dϕ + τQ /Ya 0.791 0.791 R p R p ˜ Corporate business income ydϕ + pxI dϕ + τQC /Ya 0.608 0.608 y+pxI ≥v¯ y+pxI ≥v¯ Corporate labor income wLC /Ya 0.385 0.385

Net operating surplus NOS/Ya 0.100 0.124 R p Consumption of ﬁxed capital δT kT dϕ /Ya 0.075 0.051 y+pxI ≥v¯ Production taxes τQC /total income 0.048 0.048 R p R p ˜ Noncorporate business income ydϕ + pxI dϕ + τQ /Ya 0.183 0.183 y+pxI

Private Consumption Ch/Ya 0.612 0.632

Private Investment (KT − (1 − δT ) KT + PXI ) /Ya 0.182 0.144 Gov’t Investment + consumption G/total¯ income 0.206 0.219

Table 9: National Account Shares, Data and Benchmark Model

Then corporate business income/total business income is given by:

P Y + P X + τQ Y + PX + τQ˜ Y C I IC C = C IC C Yb + Y¯nb 1 Yb + Y¯nb PY | {z } Ya

The national accounts of this economy can be summarized in Table9.

A.2 Firm-level Data

For measurement of intangible capital at the firm-level, I use Compustat North America- Capital IQ, which provides annual accounting data for publicly listed U.S. firms. This data set fits our purpose well because firm-level R&D investment data are available and because it is well-suited to study U.S. firmsâ financial aspects due to its rich firm characteristics and industry information. I exclude foreign firms, government-sponsored firms, public utilities and financial firms, as is commonly done in the investment literature. I also exclude mergers, acquisitions, and observations with extreme values. To be consistent with the aggregate data, I focus on the period 1975 - 2016. After following the standard data cleaning procedures (see, for example, Imrohoroglu and Tuzel(2014); Ottonello and Winberry(2018)), I end up with 14734 firms in total and 153,505 firm-year observations. The representativeness of the dataset is fairly good: total assets Compustat/Flow of Funds (non-financial corporate sector) ranges

47 from 50 to 75%. Again, since I focus on all the employer firms in my model, so I filter a subset of firms in my model based on firm size with the criterion specified in the main text to target moments constructed using Compustat data.

A.2.1 Measurement of Intangible Capital at the Firm-level

Existing literature attempts to measure intangible capital either directly or indirectly. The indirect approach is to construct a proxy using aggregate stock market or national accounting data (e.g. Karabarbounis and Neiman(2018); McGrattan and Prescott(2010b)). These approaches measure intangibles as unexplained (by physical capital) residuals of stock market value or firm productivity. The other approach is to construct aggregate measures of the different components of intangible capital directly (Corrado, Hulten, and Sichel(2005)) using a wide range of aggregate datasets including NIPA, the Services Annual Survey (SAS), surveys of employer-provided training from BLS. The aggregate data on intangibles considered in this paper is measured following the method developed by Corrado, Hulten, and Sichel(2005), henceforth, CHS. The biggest advantage of this method is that it includes the most types of intangible assets including the very firm-specific human and structural resources, in addition to software, R&D, and artistic originals which have been included into the national accounts. In general, CHS’s method includes three categories of business intangibles: (1) computerized information, which a firm’s knowledge embedded in the computer programs and computerized databases; (2) innovative property, which is a firm’s knowledge acquired through scientific R&D and non-scientific inventive and creative activities; (3) economic competencies, which is a firm’s knowledge embedded in firm-specific human and structural resources, including brand equity and on-the-job training. For the purpose of accounting for the BEA-measured income shares, I target the BEA- measured intangibles (or IPP called by the BEA) on the aggregate. To be consistent with the BEA method on the firm-level, I construct a measure of intangible capital including software and R&D using Compustat data. The difficulty is that the capital that is created by investments in intangible assets such as R&D are only expensed, thus not being reported on firms’ balance sheets. Following the method developed by Peters and Taylor(2016) and Falato et al.(2018), the essential idea to overcome this difficulty is to capitalize expenses related to intangible assets consistent with BEA.

48 More specifically, the replacement cost of knowledge capital is measured by capitalizing R&D expenditures using perpetual inventory method with depreciation rate of 20%. Since firm-level expenses on software are not available, I construct an industry-level measure to approximate the intangible assets on them. The BEA classification features 63 industries. I match the BEA data to Compustat firm-level data using SIC codes, assuming that, for a given year, firms in the same industry have the same shares of intangible assets on software. I construct measures of software shares for industry l in year t as

BEA BEA softwaret IPPl,t × BEA IPPt Compustat softwarel,t = BEA F ixedAssetl,t F ixedAssetl,t

Compustat where F ixedAssetl,t are total assets in industry l in year t. Since data on the fixed assets of software are only available at the aggregate level rather than the industry level but intellectual property products are, I use the economy-wide ratio of software to IPP multiplied by the IPP at the industry level to approximate the industry-level intangible assets on software. The BEA data comes from Fixed Assets Accounts Tables (FAT) Table 3.7I. Informational capital is constructed by capitalizing expenditures on software with a depreciation rate of 31% following BEA. Knowledge capital (in terms of the replacement cost), information capital, and the on-the- balance-sheet intangibles consists of the intangible capital stock on the firm-level. When a firm purchases an intangible asset externally, for example, by acquiring another firm, the firm typically capitalizes the asset on the balance sheet as part of Intangible Assets, which equals the sum of Goodwill and Other Intangible Assets. The asset is booked in Other Intangible Assets if the acquired asset is separately identifiable, such as a patent, software, or client list60. Acquired assets that are not separately identifiable, such as human capital, are in Goodwill. When an intangible asset becomes impaired, firms are required to write down its book value. There is debate about whether on-the-balance sheet "intangibles" should be added into the measure of intangible capital on the firm-level or not. Following Peters and Taylor(2016), I keep Goodwill in Intangible Assets in my main analysis, because Goodwill does include the fair cost of acquiring intangible assets that are not separately identifiable. However, it should be noted that due to the inclusion of goodwill, the item picks up over-payment in mergers

60But this variable "intangibles-Other" variable in Compustat that is net of goodwill is only available in Compustat since 2000

49 coeﬃcient estimate Data Model α -0.078*** -0.043*** ***p < 0.01 Model simulated starting with entrants distribution for 50,000 ﬁrms and 100 years

Table 10: Financing patterns

Dependent variable: market share (A)(B)(C) Model Compustat intangible share 0.105*** 0.025*** 0.014*** 0.258*** Industry×Year F.E.. Yes No No - Firm F.E. No Yes Yes - Year F.E. No No Yes - ***p < 0.01 Model simulated starting with entrants distribution for 50,000 ﬁrms and 100 years

Table 11: Intangibles and market share

& acquisitions (M&As). Hence, I also try excluding Goodwill from external intangibles as a robust check. In general, my resulting estimate for the ratio of intangible to tangible capital over the past decades is comparable to the estimate based on BEA. Next, I discuss two empirical patterns I generate from Compustat and compare the corresponding patterns generated from my model with them, which serve as validations. First, I find that intangible-intensive firms tend to rely less on external debt finance. More specifically, I run the following OLS regression:

Net Debtit Intangible Capitalit = α + βXit + Y eardummy + Industrydummy + εit Assetit Assetit where X: a set of control variables including Tobin’s Q, physical investment/assets ratio, firm size, dummy variables for positive dividend payout. I report the results in Table 10 with comparison between model and data. Second, I find that a firm’s market share in its industry is higher when its intangible capital to total assets is higher, and this relationship holds between firms of the same industry, within firms over time, and controlling for year effects. This is in line with Crouzet and Eberly (2019) although they measure intangible capital at the firm-level different from mine 61.I

61Crouzet and Eberly(2019) don’t capitalize R&D expenses and software expenses. That is, they only

50 Data Source Number of Employees BDS 2007 119,627,020 SBO PUMS 2007 30,465,820 Compustat 2007 73,434,000 BEA 2007 118,944,000 (all private domestic industries) Employer Firms Employer Firms’ receipts Nonemployer Firms Nonemployer Firms’ receipts Data Source (number) ($1000) (number) ($1000) SBO 2007 5,287,344 10,015,051,752 21,104,893 932,487,829 BDS 2007 5,240,019 - 0 0

Table 12: Coverage of Firms in Multiple Datasets Note: The SBO covers all nonfarm businesses ﬁlling IRS tax forms as individual proprietorships, partnerships, or any type of corporation with receipts of $1,000 or more. However, businesses classiﬁed in the SBO as publicly owned are not included in the PUMS version. report results in Table 11.

A.2.2 Other Firm-level Moments

For firm distribution, I focus on all the employer firms in U.S. business sector. For statistics related to firm entry, exit, job creation and destruction, firm size and age distribution, I calculate directly from the Business Dynamic Statistics (BDS), which are compiled from the Longitudinal Business Database (LBD). The LBD is a confidential longitudinal database of business establishments and firms starting from 1976. Any moments I need to rely on LBD, I draw them from the existing literature with corresponding years. As I’ve mentioned in the main text, I admit that there exists discrepancy between the coverage of BEA-business sector, which covers both employer and non-employer firms and the coverage of BDS/LBD, which only covers employer firms. However, as shown in Table 12, non-employer firms do not contribute to the total employment and only takes a very small portion of total sales (which is 2.48% based on SBO 2007), I assume BEA-business sector and BDS/LBD cover the same firms.

consider intangibles on the balance sheet.

51 Change in Data (pp.) Model (pp.) Category Labor share (post-2013-revision) -4.6 -2.7 Labor share (pre-1999-revision) -2.4 -1.1 Saving flows/GVA +6.0 +5.0 Annual Firm Entry rate -4.5 -2.9 Employment Share of Large Firms (500+) +4.7 +4.2 Employment Share of Mature Firms (11+) +13.7 +6.9 Top 10% concentration (Sales) +5.3 +4.4 Average firm size +12.2 +8.9 Labor productivity dispersion +8.0 +5.5 Labor productivity gap +20.00 +7.20 between frontier and lagged firms

Table 13: Aggregate Implications: Additional Results

Appendix B: Empirical Facts

In this section, I summarize the empirical facts, mainly declined labor share and increased concentration, that my paper attempts to explain, together with some additional facts regarding business dynamism that my model can account for as well. To summarize, I focus on the following regularities, most of which occur during the the past three decades:

1. The labor share of gross value-added has gone down.

2. The employment share of large ﬁrms (500+ employees) has risen.

3. The employment share of mature ﬁrms (11+ years) has risen.

4. Market concentration in terms of top 10% has risen.

5. Firm entry rate has declined.

6. Productivity dispersion of ﬁrms has risen. Similarly, the labor productivity gap between frontier and laggard ﬁrms has widened.

7. The corporate saving ﬂows relative to gross value-added has risen, together with weaker physical investment.

I summarize the results on how my model accounts for the additional empirical facts driven by the IISTC besides labor share and concentration in Table 13.

52 Figure 6: Productivity gap between frontier and laggard ﬁrms Source: Andrews, Criscuolo, and Gal(2015)

Figure 7: Labor productivity dispersion Source: Decker et al.(2018)

53 Figure 8: Corporate saving ﬂows & Physical Investment Source: Flow of Funds; BEA-NIPA

Appendix C: Computational details

I essentially follow an algorithm that is similar to Gavazza, Mongey, and Violante(2018) and solve value and policy functions using a method developed by Mongey(2015) 62.

C.1 Algorithm

I describe the solution method for my long-run GE economy in this section. The usual nested fixed point approach is extended in order to accommodate the additional features of my model. That is, the essence of our approach is to guess a set of prices, compute decision rules (given prices) to simulate the economy, and finally verify whether those are the equilibrium prices. Specifically, I execute the following steps:

1. Make an initial guess for the wage w˜. With specifying the upper bound and lower u l u l w0 +w0 bound for wage, w0 , w0, the initial guess can be 2 . Due to the risk neutrality of

62This method involves programming in Matlab with MEX, which relies on the COMPECON toolbox developed by Miranda and Fackler(2002). The toolbox can be found at http://www4.ncsu.edu/˜pfackler/compecon/toolbox.html

54 1 households, the real interest rate is r = β − 1 at the steady state.

2. Solve both incumbent firms’ and entrants’ dynamic programming problem described in Section 3 at the prices w,˜ r. The general procedure involves using a nested vectorized golden search method to solve the optimization problems and find the policy functions 0 0 kI (s) , a (s) . More specifically, we solve the optimization problem in two steps: (1) 0 0 solve the policy function for a (s) given kI (s). We first solve (kT 1, l1) which is the solution to a static problem of firms. We then define a cash-in-hand variable cih =

y − (r + δT )kT 1 − wl˜ 1 + (1 + r) a − κo, and solve kT 2, l2 according to the production 0 technology for intangible investment goods (X) given kI , zI , and prices RT = r + δT and

w˜. Next, check two things: whether cih − wl2 − RT kT 2 ≥ 0 or not and whether kT < λa

or not where kT = kT 1 + kT 2. Depending on the answers to the two checking questions,

we are going to discuss three cases. Case I: cih − wl2 < 0. In this case, we simply set 0 a = 0 and d = cih − wl2 − RT kT 2. Negative dividend payment serves as a penalty.

Case II: cih − wl2 − RT kT 2 ≥ 0 and kT = λa. In this case, ﬁrms are constrained, thus 0 paying zero dividend, i.e. d = 0. Then a = min (cih − wl2 − RT kT 2, amax). Case III:

cih − wl2 − RT kT 2 ≥ 0 and kT < λa. In this case, ﬁrms are NOT constrained and thus 0 0 need to pay the dividend. Use golden search to solve for a , d . (2) after expressing a 0 0 in terms of kI , solve for kI using golden search again. More details about the method I used to solve the policy functions will be discussed in section C.2.

3. Using the policy functions for both incumbents and entrants to ﬁnd the stationary distribution using the method discussed in section C.3.

4. Given the stationary distribution and policy functions of ﬁrms, compute the aggregate labor demand LD.

5. Compare the aggregate labor demand LD at prices w,˜ r with the inelastic labor supply N¯. If |LD − N¯| < tolerance, then we are done. Otherwise, we employ a bisection price updating scheme. More speciﬁcally, we update the guess for wage based on the following rule: if LD > N¯, set the lower bound equal to the lower bound the same l l as before, w = w old and the upper bound equal to the wage in the current iteration, u ¯ l u u wu+wl w =w ˜; if LD < N, set w =w ˜ and w = w old. Then a new guess for wage is 2 and return to step 2.

55 C.2 Value functions and policy functions

I use collocation methods to solve the firm’s value function problem (3), (4), (5), and (6). Let s ≡ (kI , a, z) be the firm’s idiosyncratic state, abstracting from heterogeneity in (η, zI ) which e 0 0 0 are both permanent. I solve for an approximant of the expected value function V kI , a , z which gives the firm’s expected value conditional on current decisions for intangible capital, tangible assets, and productivity:

Z e 0 0 0 0 0 0 0 V kI , a , z = V kI , a , z dΓ z, z Z where the integrand is the value given in (4).

I set up a grid of collocation nodes S = KI × A × Z with NkI = Na = Nz = 10. Following Gavazza, Mongey, and Violante (2018), I construct Z by ﬁrst creating equi-spaced nodes from 0.001 to 0.999, which I then invert through the cumulative distribution function of the stationary distribution implied by the AR(1) process for z to obtain Z. This ensures better

coverage in the higher probability regions for z. I also choose KI ,A to have a higher density ¯ at lower values. The upper bound for intangible capital, kI , is chosen so that the optimal ∗ ¯ size of the highest productivity firm kI (¯z) is less than kI . I choose the upper bound for ∗ tangible assets, a¯, so that the maximum optimal physical capital kT 1 (¯z) can be financed, that ∗ is, kT 1 (¯z) ≤ λa¯. Note that KI ,A, and Z are parameter dependent, and therefore recomputed for each new vector of parameters considered in estimation. I approximate the expectation of value function V e (s) on S using a linear spline63 with e NkI × Na × Nz coefficients. Given a guess for the spline’s coefficients cj, cj ,I j=1,··· ,Ns iterate towards a vector of coefficients that solve the system of Ns Bellman equations, which are linear in the Ns unknown coefficients. More specifically, I have:

Ns Ns X e X e V (sj) = φ (sj) cj,V (sj) = φ (sj) cj j=1 j=1

where φ (·) is a basis function. Each iteration proceeds as follows. Given the spline coeﬃcients, I use vectorized golden search to compute the optimal policies for all states s ∈ S and the value function V (s). I then

63Alternatively, you may use cubic spline or Chebyshev polinomials.

56 ﬁt another spline to V (s), which facilitates the integration of productivity shocks ε ∼ N (0, σz). To compute V e (s) on S, I approximate the integral by

Nε e X e V (kI , a, z) = wiV (kI , a, exp (ρzlog (z) + εi)) i=1

Here, Nε = 80, and the values of εi are constructed by creating a grid of equi-spaced nodes between 0.001 to 0.999, then using the inverse cumulative distribution function of the shocks

(normal) to create a grid in ε. The weights wi are given by the probability mass of the normal distribution centered on each εi. The major advantages of having continuously distributed productivity shocks rather than discretizing it by employing Tauchen or Rouwenhorst are threefold. First, I can compute the integral more precisely. Second, in a model that features endogenous entry and exit, continuously distributed productivity shocks ensures much easier convergence in the value function iteration, compared to discretization. Another important advantage is that by having continuously distributed productivity shocks, you can have a sufficiently dense grid of productivities for the potential entrants, while in the case of discretization, the number of grids of productivities must be equal to Nz = 10. When the number of grids of productivities is too small, changing in prices may lead to either too much entry or too little entry, which makes it much harder to find the equilibrium. Having a sufficiently dense grid of productivities is also important to calibration since it ensures that entry does not jump across parameters.

C.3 Stationary distribution

To construct the stationary distribution, as Gavazza, Mongey, and Violante(2018), I use the method of non-stochastic simulation from Young(2010), modiﬁed to accommodate a continuously distributed stochastic state. I create a new, ﬁne grid of points Sf on which I approximate the stationary distribution using a histogram, setting NkI f = Naf = Nzf = 100. Given 0 f our approximation of the expected continuation value, I solve for the policy functions kI s 0 f and a s on the new grid and use these to create two transition matrices QkI , Qa which f f determine how mass shifts from points s ∈ S to points in KI f, Af, respectively. I construct

Qx as follows for x ∈ {kI , a}:

57  0 f f f 0 f  x si − Xj Xj+1 − x si h i h i Qx [i, j] = 1 0 f f f f f + 1 0 f f f f f  , x si ∈ Xj−1,Xj x si ∈ Xj ,Xj+1 Xj − Xj−1 Xj+1 − Xj

f f for i = 1,...,Ns and j = 1,...,Nx . The transition matrix for the process for z is computed

PNε i i 0 f by Qz = i=1 wiQz, where Qz is computed as above under z s = exp (ρzlog (z) + εi).

The overall incumbent transition matrix Q is simply the tensor product Q = Qz ⊗ Qa ⊗ QkI . To compute the stationary distribution, I still need the distribution of entrants. To allow for entry cutoﬀs to move smoothly, I compute entrant policies on a dense grid of Nz0 = 100 productivities. The grid Z0 is constructed by taking an equally spaced grid in [0.01, 0.99] and inverting it through the cumulative distribution function of potential entrant productivities

(exponential). Let the corresponding vector of weights be given by P0. Given the approxima- e tion of the continuation value V (s0) , I can solve the potential entrant’s policies conditional on entry. I can then solve the ﬁrm’s discrete entry decision. Finally, I compute an equivalent transition matrix Q0 using these policies, where non-entry results in a row of zeros in Q0 . The discretized stationary distribution L on Sf is then found by the following approximation to the law of motion 0 0 L = (1 − πd) Q L + M0Q0 (P0) , which is a contraction on L , solved by iterating on a guess for L . The ﬁnal stationary f PNs distribution is found by choosing M0 such that i=1 Li = 1.