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Power-law distributions of corporate innovative output: evidence from U.S. patent data

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Abstract

This study aims to examine the existence and the characteristics of power laws in the distribution of corporate innovative output. Using a dataset containing information on 1,102,839 U.S. patent applications by 94,103 U.S. private firms during the period of 1976–2000, we find that corporate innovative output, as measured by either simple or quality-adjusted patent counts, follows power-law distributions that theoretically have infinite variances and, in some cases, infinite means. In addition, we find that corporate innovative output is power-law distributed in all technological fields. We further find that the power-law distribution of corporate innovative output tends to be stronger (i.e., the concentration of corporate innovative output in a few firms is more pronounced) in technological fields with more abundant technological opportunities and for firms with higher technological competence.

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Notes

  1. For more examples, see, among others, Newman (2005), Clauset et al. (2009), and Gabaix (2016).

  2. Furthermore, power-law-distributed data are scarce compared to those following other heavy-tailed distributions such as lognormal, the Weibull, and the exponentially truncated power-law distributions (Broido and Clauset 2019). Voitalov et al. (2018) even argued that a broader definition of a power-law distribution including a pure power-law distribution and its variants is necessary as data that follow a (pure) power-law distribution are rare.

  3. As a result, the annual number of patent applications in the NBER data file sharply declines since 2000.

  4. In addition to the number of forward citations, there are other indicators of patent quality such as the number of patent renewal (Schankerman and Pakes 1986), the frequency of patent litigation (Lanjouw and Schankerman 2001), and family sizes (Harhoff et al. 2003a). It is also worth noting that the quality of patents can be measured by combinations of their intrinsic characteristics such as the numbers of backward citations, self-citations, claims, and co-inventors (Lanjouw and Schankerman 2004; Higham et al. 2019).

  5. According to the NBER data file, the average number of citations made by each patent increased gradually, which may be, as Hall et al. (2001) conjectured, attributed to changes in patent examination practices. In addition, the average number of citations either made or received by patents largely varies according to technological fields.

  6. It is also worth noting that the number of forward citations of patents can also be adjusted by their application years using Higham et al. (2017)’s method to estimate the effect of patents’ aging on their number of forward citations.

  7. The RTA index can be interpreted as the ratio of firm i‘s share of patent applications within technological field j (\(\frac{{N_{i,j} }}{{\mathop \sum \nolimits_{i} N_{i,j} }}\)) to the firm‘s share of patent applications across the entire technological fields (\(\frac{{\mathop \sum \nolimits_{j} N_{i,j} }}{{\mathop \sum \nolimits_{i} \mathop \sum \nolimits_{j} N_{i,j} }}\)). It also means the relative concentration of firm i‘s patent applications in technological field j (\(\frac{{N_{i,j} }}{{\mathop \sum \nolimits_{j} N_{i,j} }}\)) compared to the concentration of all firms’ patent applications in the technological field (\(\frac{{\mathop \sum \nolimits_{i} N_{i,j} }}{{\mathop \sum \nolimits_{i} \mathop \sum \nolimits_{j} N_{i,j} }}\)). Either interpretation indicates that the RTA index represents firm i’s comparative advantage in technological field j (Patel and Pavitt 1997).

  8. The lower bound can be estimated by choosing xmin that yields the best goodness of fit under the assumption that observations smaller than xmin follow an exponential distribution, whereas those greater than xmin follow a power-law distribution (Clauset et al. 2009).

  9. The one-sample Kolmogorov–Smirnov test allows us to test the goodness of fit of a power-law distribution without any alternative distributions (Clauset et al. 2009). However, this method frequently fails when data have any noise, imperfections, or correlations (Alstott et al. 2014; Gerlach and Altmann 2019). For these reasons, we choose the approach of using an exponential distribution as an alternative distribution.

  10. On the contrary, the scaling parameter of the power-law distribution of QPC or C_QPC is smaller (greater) than that of the power-law distribution of PC when the number of high-quality patents filed by each firm tends to increase more (less) than proportionally with the number of overall patent applications by each firm. A more detailed explanation is given in “The relationship between simple patent counts (PC) and Quality-adjusted patent counts (QPC and C_QPC)” of Appendix.

  11. Throughout this study, patent applications filed during the periods of 1976–1980, 1981–1985, 1986–1990, 1991–1995, and 1996–2000 are denoted by Year Cohort 1, Year Cohort 2, Year Cohort 3, Year Cohort 4, and Year Cohort 5, respectively.

  12. Note that scaling parameters for some of the subsamples classified by the year of patent applications are significantly greater than those for the entire sample (year 2000 for PC and years 1986, 1987, 1990, 1991, 1993 for QPC and C_QPC). one plausible explanation is that the concentration of innovative output in a few firms is more pronounced for a longer period due to the cumulativeness or the positive feedback effect associated with the production of innovative output (e.g. Griliches 1986; Romer 1990; Jones 1995). In addition, given that the scaling parameter of a power-law distribution is sensitive to sample sizes (e.g., Rosen and Resnick 1980; Guérin-Pace 1995), temporal fluctuations in the scaling parameter can often occur.

  13. The two-sample Kolmogorov–Smirnov test checks whether two different data samples are drawn from the same distribution. Note that the statistically significant statistics indicate that the two data samples follow different distributions.

  14. The results using PC and QPC are largely consistent with our findings, as shown in “Power-law distributions of corporate innovative output by technological opportunity (PC)” and “Power-law distributions of corporate innovative output by technological opportunity (QPC)” of Appendix.

  15. The results using PC and QPC are largely consistent with our findings, as shown in “Power-law distributions of corporate innovative output by firm-specific technological competence (PC)” and “Power-law distributions of corporate innovative output by firm-specific technological competence (QPC)” of Appendix. It is worth noting that the goodness of fit of the power-law distribution of QPC for Year Cohort 4 (1991–1995) is even statistically insignificant for firms with lower technological competence, while it is statistically significant for firms with higher technological competence. This result shows that the power-law distribution of corporate innovative output tends to be stronger (i.e., the concentration of innovative output in a few firms is more pronounced) for firms with higher technological competence in terms of not only the scaling parameter but also the goodness of fit.

  16. The difference between the mechanisms is that the power-law-generating mechanism is accompanied by additional conditions such as a positive random walk (Kesten 1973), a reflecting barrier (e.g., Gabaix 1999), or reset events (e.g., Reed 2001).

  17. The 6 sections and 23 subsections of the IPC are listed in “Technological categories of the international patent classification” of Appendix.

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  1. College of Business, Korea Advanced Institute of Science and Technology (KAIST), 85 Hoegi-ro, Dongdaemun-gu, Seoul, 02455, Republic of Korea

    Mincheol Choi

  2. College of Business, KAIST, Seoul, Republic of Korea

    Chang-Yang Lee

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Appendix

Appendix

The relationship between simple patent counts (PC) and quality-adjusted patent counts (QPC and C_QPC)

For simplicity, let us express the relationship between PC and A_PC (A_PC can be either QPC or C_QPC) as \({\mathbf{A}}\_{\mathbf{PC}} \propto {\mathbf{PC}}^{\beta }\), where β is an arbitrary positive number. If PC follows a power-law distribution with the scaling parameter of αPC (i.e., (\(\Pr ({\mathbf{PC}} > x_{\text{PC}} ) \propto x_{\text{PC}}^{{ - \alpha_{\text{PC}} }}\)), then A_PC also follows a power-law distribution, as shown below:

$$\Pr \left( {{\mathbf{A}}\_{\mathbf{PC}} > x_{{{\text{A}}\_{\text{PC}}}} } \right) = \Pr ({\mathbf{PC}} > x_{\text{PC}} ) \propto x_{\text{PC}}^{{ - \alpha_{\text{PC}} }} \propto x_{{{\text{A}}\_{\text{PC}}}}^{{ - \alpha_{\text{PC}} /\beta }} .$$
(11)

As shown in Equation (11), similar (not statistically significantly different) scaling parameters of the power-law distributions of PC (αPC) and A_PC (αA_PC = αPC/β) mean that the relationship between PC and A_PC is roughly proportional (i.e., \(\beta \cong 1\)). On the contrary, αA_PC is smaller than αPC when A_PC tends to increase more than proportionally with PC (i.e., β > 1), while αA_PC is greater than αPC when A_PC tends to increase less than proportionally with PC (i.e., β < 1).

Corporate innovative output distributions by technological fields (one-digit technological classification by Hall et al. (2001)): 1976–2000

Samples

Obs.

PC

QPC

C_QPC

Cat. 1

18,946

0.83 ± 0.04 (13.47)***

0.83 ± 0.03 (15.03)***

0.82 ± 0.03 (15.39)***

Cat. 2

16,989

0.90 ± 0.03 (9.43)***

0.92 ± 0.04 (7.63)***

0.92 ± 0.05 (7.40)***

Cat. 3

12,214

0.88 ± 0.05 (12.85)***

0.82 ± 0.04 (11.45)***

0.81 ± 0.04 (11.35)***

Cat. 4

19,263

0.92 ± 0.04 (10.67)***

0.93 ± 0.05 (9.93)***

0.91 ± 0.05 (9.54)***

Cat. 5

28,866

1.01 ± 0.03 (13.13)***

1.03 ± 0.06 (10.51)***

1.02 ± 0.05 (11.08)***

Cat. 6

37,938

1.06 ± 0.05 (14.24)***

1.02 ± 0.03 (14.71)***

1.01 ± 0.03 (15.67)***

  1. The 99% confidence intervals of the scaling parameters are (αMLE) presented
  2. The log-likelihood ratios are in the parentheses
  3. ***, **, * denote statistical significance at the 1%, 5%, and 10% levels, respectively

Technological categories of the International Patent Classification

Section

Subsection

A: Human necessities

A0: Agriculture

A2: Foodstuffs; tobacco

A4: Personal or domestic articles

A6: Health; amusement

B: Performing operations; transporting

B0: Separating; mixing

B2: Shaping

B4: Printing

B6: Transporting

B8: Micro-structural technology; nano-technology

C: Chemistry; metallurgy

C0: Chemistry

C2: Metallurgy

C4: Combinatorial technology

D: Textiles; paper

D0: Textiles or flexible materials not otherwise provided

D2: Paper

E: Fixed constructions

E0: Building

E2: Earth or rock drilling; mining

F: Mechanical engineering; lighting; heating; weapons; blasting

F0: Engines or pumps

F1: Engineering in general

F2: Lighting; heating

F4: Weapons; blasting

G: Physics

G0: Instruments

G2: Nucleonics

H: Electricity

H0: Electricity

  1. The technological fields with less than 1000 patents (B8 and C4) are not analyzed in this study

Corporate innovative output distributions by technological fields (one-digit International Patent Classification): 1976–2000

Samples

Obs.

PC

QPC

C_QPC

A

25,717

0.98 ± 0.04 (13.16)***

0.87 ± 0.03 (12.86)***

0.88 ± 0.03 (11.86)***

B

33,893

1.02 ± 0.05 (12.84)***

0.98 ± 0.03 (14.29)***

0.98 ± 0.04 (12.44)***

C

11,094

0.74 ± 0.03 (16.26)***

0.74 ± 0.03 (15.13)***

0.76 ± 0.04 (12.89)***

D

2079

0.91 ± 0.10 (6.07)***

0.91 ± 0.11 (5.02)***

0.94 ± 0.15 (4.02)***

E

8264

1.08 ± 0.07 (8.45)***

1.07 ± 0.08 (6.87)***

1.06 ± 0.08 (6.64)***

F

15,601

1.02 ± 0.05 (8.99)***

1.04 ± 0.06 (7.17)***

1.09 ± 0.09 (6.15)***

G

25,005

0.94 ± 0.04 (8.75)***

0.96 ± 0.04 (7.68)***

0.96 ± 0.04 (7.53)***

H

16,159

0.87 ± 0.03 (11.59)***

0.91 ± 0.07 (7.92)***

0.86 ± 0.03 (10.09)***

  1. The 99% confidence intervals of the scaling parameters are (αMLE) presented
  2. The log-likelihood ratios are in the parentheses
  3. ***, **, * denote statistical significance at the 1%, 5%, and 10% levels, respectively

Corporate innovative output distributions by technological fields (two-digit International Patent Classification): 1976–2000

Samples

Obs.

PC

QPC

C_QPC

A0

4255

0.99 ± 0.08 (7.28)***

0.99 ± 0.07 (7.30)***

0.98 ± 0.07 (7.44)***

A2

2286

1.00 ± 0.11 (6.97)***

0.96 ± 0.09 (7.10)***

0.94 ± 0.08 (7.24)***

A4

8010

1.31 ± 0.12 (7.09)***

1.23 ± 0.10 (7.26)***

1.12 ± 0.06 (9.88)***

A6

14,571

0.95 ± 0.05 (11.44)***

0.84 ± 0.04 (11.32)***

0.84 ± 0.04 (10.43)***

B0

9911

1.02 ± 0.06 (10.38)***

1.03 ± 0.09 (9.69)***

1.02 ± 0.07 (10.79)***

B2

14,195

1.00 ± 0.05 (10.44)***

0.93 ± 0.40 (11.05)***

0.91 ± 0.04 (10.87)***

B4

26,04

1.01 ± 0.14 (4.50)***

1.07 ± 0.09 (4.95)***

1.03 ± 0.10 (4.39)***

B6

16,168

1.01 ± 0.04 (11.62)***

1.03 ± 0.05 (10.64)***

1.02 ± 0.05 (11.28)***

C0

9490

0.73 ± 0.03 (15.78)***

0.72 ± 0.03 (14.70)***

0.75 ± . 0.04 (12.72)***

C2

2731

0.95 ± 0.09 (6.03)***

0.84 ± 0.07 (5.80)***

0.85 ± 0.07 (4.82)***

D0

1786

0.96 ± 0.11 (5.32)***

1.01 ± 0.18 (4.86)***

1.01 ± 0.17 (4.71)***

D2

456

0.82 ± 0.14 (4.85)***

0.86 ± 0.21 (2.74)***

0.85 ± 0.19 (3.22)***

E0

6933

1.17 ± 0.08 (6.12)***

1.30 ± 0.13 (4.58)***

1.10 ± 0.06 (8.84)***

E2

1645

0.80 ± 0.15 (6.80)***

0.79 ± 0.09 (7.52)***

0.78 ± 0.09 (7.07)***

F0

3701

0.92 ± 0.08 (6.49)***

0.91 ± 0.06 (6.98)***

0.89 ± 0.06 (7.01)***

F1

7703

0.99 ± 0.06 (7.73)***

0.98 ± 0.05 (8.12)***

0.94 ± 0.04 (8.67)***

F2

6105

1.16 ± 0.09 (6.58)***

1.12 ± 0.08 (7.06)***

1.10 ± 0.07 (7.32)***

F4

1137

0.98 ± 0.12 (5.58)***

1.06 ± 0.14 (5.43)***

1.05 ± 0.12 (6.71)***

G0

24,755

0.94 ± 0.03 (9.01)***

0.96 ± 0.04 (7.61)***

0.95 ± 0.04 (7.60)***

G2

629

1.03 ± 0.17 (3.31)***

1.04 ± 0.19 (3.00)***

1.18 ± 0.31 (3.00)***

H0

16,159

0.87 ± 0.03 (11.59)***

0.91 ± 0.07 (7.92)***

0.86 ± 0.03 (10.09)***

  1. The 99% confidence intervals of the scaling parameters are (αMLE) presented
  2. The log-likelihood ratios are in the parentheses
  3. ***, **, * denote statistical significance at the 1%, 5%, and 10% levels, respectively

Power-law distributions of corporate innovative output by technological opportunity (PC)

Variable: PC

Technological opportunity (OPP)

High-opportunity technological fields

Low-opportunity technological fields

Two-sample K–S statistics

Obs.

αMLE (log-likelihood ratio)

Obs.

αMLE (log-likelihood ratio)

Entire sample (1976–2000)

72,618

0.93 ± 0.02 (17.29)***

39,921

1.00 ± 0.05 (10.54)***

0.45***

Year cohort 1 (1976–1980)

14,442

0.83 ± 0.06 (12.20)***

6672

0.98 ± 0.06 (8.44)***

0.64***

Year cohort 2 (1981–1985)

15,471

0.89 ± 0.05 (13.78)***

6966

1.01 ± 0.08 (7.75)***

0.31***

Year cohort 3 (1986–1990)

20,591

0.93 ± 0.05 (12.18)***

9810

1.02 ± 0.09 (7.03)***

0.15***

Year cohort 4 (1991–1995)

25,143

0.92 ± 0.04 (11.06)***

13,469

1.08 ± 0.05 (10.18)***

0.57***

Year cohort 5 (1996–2000)

31,740

0.91 ± 0.04 (9.74)***

16,598

1.07 ± 0.05 (10.67)***

0.49***

  1. The 99% confidence intervals of the scaling parameters (αMLE) are presented
  2. The log-likelihood ratios are in the parentheses
  3. ***, **, * denote statistical significance at the 1%, 5%, and 10% levels, respectively

Power-law distributions of corporate innovative output by technological opportunity (QPC)

Variable: QPC

Technological opportunity (OPP)

High-opportunity technological fields

Low-opportunity technological fields

Two-sample K–S statistics

Obs.

αMLE (log-likelihood ratio)

Obs.

αMLE (log-likelihood ratio)

Entire sample (1976–2000)

72,618

0.90 ± 0.03 (13.05)***

39,921

1.01 ± 0.04 (10.92)***

0.16***

Year cohort 1 (1976–1980)

14,442

0.90 ± 0.03 (16.27)***

6672

0.99 ± 0.06 (7.92)***

0.26***

Year cohort 2 (1981–1985)

15,471

0.93 ± 0.04 (13.14)***

6966

1.05 ± 0.09 (5.79)***

0.19***

Year cohort 3 (1986–1990)

20,591

0.97 ± 0.06 (9.67)***

9810

1.00 ± 0.06 (7.21)***

0.67***

Year cohort 4 (1991–1995)

25,143

0.85 ± 0.03 (11.05)***

13,469

1.10 ± 0.10 (7.13)***

0.61***

Year cohort 5 (1996–2000)

31,740

0.93 ± 0.04 (8.20)***

16,598

1.02 ± 0.06 (9.88)***

0.55***

  1. The 99% confidence intervals of the scaling parameters (αMLE) are presented
  2. The log-likelihood ratios are in the parentheses
  3. ***, **, * denote statistical significance at the 1%, 5%, and 10% levels, respectively

Power-law distributions of corporate innovative output by firm-specific technological competence (PC)

Variable: PC

Firm-specific technological competence (CORE)

High-competence firms

Low-competence firms

Two-sample K–S statistics

Obs.

αMLE (log-likelihood ratio)

Obs.

αMLE (log-likelihood ratio)

Entire sample (1976–2000)

46,926

0.92 ± 0.03 (15.60)***

47,177

2.56 ± 0.24 (3.07)***

0.49***

Year cohort 1 (1976–1980)

8969

0.79 ± 0.03 (16.81)***

8969

1.90 ± 0.17 (3.53)***

0.22***

Year cohort 2 (1981–1985)

9347

0.84 ± 0.04 (14.88)***

9805

2.24 ± 0.31 (2.83)***

0.27***

Year cohort 3 (1986–1990)

13,054

0.89 ± 0.06 (11.04)***

13,075

1.96 ± 0.21 (4.10)***

0.69***

Year cohort 4 (1991–1995)

16,330

0.89 ± 0.04 (11.96)***

16,549

2.39 ± 0.29 (2.76)***

0.31***

Year cohort 5 (1996–2000)

20,544

0.89 ± 0.03 (11.20)***

20,559

2.21 ± 0.18 (2.58)***

0.43***

  1. The 99% confidence intervals of the scaling parameters (αMLE) are presented
  2. The log-likelihood ratios are in the parentheses
  3. ***, **, * denote statistical significance at the 1%, 5%, and 10% levels, respectively

Power-law distributions of corporate innovative output by firm-specific technological competence (QPC)

Variable: QPC

Firm-specific technological competence (CORE)

High-competence firms

Low-competence firms

Two-sample K–S statistics

Obs.

αMLE (log-likelihood ratio)

Obs.

αMLE (log-likelihood ratio)

Entire sample (1976–2000)

46,926

0.90 ± 0.03 (11.41)***

47,177

2.34 ± 0.25 (2.74)***

0.99***

Year cohort 1 (1976–1980)

8969

0.83 ± 0.05 (11.88)***

8969

1.83 ± 0.18 (2.96)***

0.43***

Year cohort 2 (1981–1985)

9347

0.87 ± 0.05 (10.63)***

9805

2.16 ± 0.28 (3.77)***

0.36***

Year cohort 3 (1986–1990)

13,054

0.89 ± 0.04 (9.89)***

13,075

1.77 ± 0.13 (3.99)***

0.62***

Year cohort 4 (1991–1995)

16,330

1.03 ± 0.11 (5.14)***

16,549

2.43 ± 0.47 (1.12)

0.97***

Year cohort 5 (1996–2000)

20,544

0.88 ± 0.04 (7.91)***

20,559

2.20 ± 0.28 (3.18)***

0.42***

  1. The 99% confidence intervals of the scaling parameters (αMLE) are presented
  2. The log-likelihood ratios are in the parentheses
  3. ***, **, * denote statistical significance at the 1%, 5%, and 10% levels, respectively

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Choi, M., Lee, CY. Power-law distributions of corporate innovative output: evidence from U.S. patent data. Scientometrics 122, 519–554 (2020). https://doi.org/10.1007/s11192-019-03304-8

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Keywords

JEL Classification