The added value of management skill in the explanation of the distribution of firm size


The stochastic explanations of skewed distribution of firm size question the value added of management theories that trace differences in market share and profits of firms to strategy and organization choices, which are in turn linked to better entrepreneurial skills. This paper explains the distribution of firm size as the market equilibrium outcome of individual occupational choices of working as entrepreneurs or employees. The distribution of size and profit of firms in the equilibrium is directly related to the distribution of skills within the group of individuals who, in the same equilibrium, choose to work as entrepreneurs, restoring the importance of management choice in the performance of firms. The paper highlights the importance of having a theoretical model as guidance in the interpretation of regularities observed in the distribution of firm size, and the ability to distinguish between “resembling” and “true” power laws of such distributions.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5


  1. 1.

    The earlier explanation is from Gibrat’s (1931) law of independence between growth rates of firms and their respective initial absolute size. Ijiri and Simon (1967) combine proportionate growth rates (Gibrat’s law) with a minimum scale in the entry of new firms to show that the distribution of firm size will converge to a Pareto distribution (power law distribution).

  2. 2.

    Coase (1937) defines the entrepreneur as “the person or persons that in a competitive system take the place of the price system in the direction of resources.” Penrose (1959) explicitly refers to the managers’ limited working time as determining the costs of growth and limits to the size of the firm; in organizational choice models, the number of entrepreneurs and the size of the respective production team depend on the organizational size diseconomies, similar to Penrose’s limits to the growth of the firm.

  3. 3.

    Although not considered in this study, empirical evidence (Davis and Henrekson 1999; Henrekson and Johansson 1999) shows that the distribution of firm size will also depend on market frictions and the institutional environment (taxes, employment laws, regulation of financial markets, and size of the public sector).

  4. 4.

    In a less strict sense, other distributions, which are not represented by pure power functions, are also referred to as power laws, like the “power law with exponential cut-off” or the piecewise functions consisting of two or more power functions. In this respect, Aguinis et al. (2018) use the term power law to refer to those heavy-tailed distributions where output is clearly dominated by a small group of elites and most individuals in the distribution are far to the left of the mean”. This definition of power law can accommodate distributions other than the Pareto and the Zeta distributions, the only two technically acceptable power laws. Then, the fact that a distribution resembles a power law does not necessarily imply that it is strictly a power law. In this paper, the power law is restricted to Pareto and Zeta distributions.

  5. 5.

    See also McKelvey (2004) on complexity science and entrepreneurship.

  6. 6.

    All the analysis can be easily extended to the case where firms also use other inputs (beyond labor) in production, for instance, capital input. See Medrano-Adan et al., 2015; 2019).

  7. 7.

    As complement, Appendix 2 shows the closed solution to the market equilibrium from occupational choices when the distribution of skill is Pareto.

  8. 8.

    There is extensive research comparing the goodness of fit of power law and of other distribution functions, for example lognormal and exponential ones, to actual data on the distribution of economic and non-economic variables (Clauset et al., 2009; Joo et al., 2017). The difference with the research reported here is that we compare the goodness of fit of a distribution predicted by a theoretical model with the goodness of fit of a power law.

  9. 9.

    The Ramsey Regression Equation Specification Error Test (RESET), Ramsey (1969), is a general specification test, which tests whether non-linear combinations of the fitted values help explain the dependent variable. If the null hypothesis that all coefficients of non-linear combinations of the fitted values are zero is rejected, then the model suffers from misspecification (Wooldridge, 2019; Green, 2012).

  10. 10.

    With binned data, the likelihood function can be approximated by \( \overline{L}\left(\right)\prod \limits_{i=1}^{13}{\left[F\left({L}_i\right)-F\left({L}_{i-1}\right)\right]}^{n_i} \). If the firm size distribution follows a Zipf distribution, then \( F\left({L}_i\right)=\frac{1}{\zeta \left(1+\alpha \right)}\sum \limits_{k=1}^{L_i}{k}^{-1-\alpha } \). The log-likelihood function to maximize is \( Log\left[\overset{\frown }{L}\right]=N\ln \left(\frac{1}{\zeta \left(1+\rho \right)}\right)+\sum \limits_{i=1}^{13}{n}_i\ln \left[\sum \limits_{k=1}^{Li}{k}^{-1-\rho }-\sum \limits_{k=1}^{Li-1}{k}^{-1-\rho}\right] \) and the optimal solution is \( {\hat{\rho}}_{ML}=0.501547 \).

  11. 11.

    We minimize the sum of squares of residuals in the “original” variable, number of firms,\( {\hat{u}}_i=\left({N}_i/N\right)-{\hat{F}}_s\left({L}_i\right)=\frac{N_i}{N}-\frac{1}{2-2{\hat{c}}_3}\left( Erfc\left[\frac{\hat{c}-\ln \left({L}_i\right)}{{\hat{c}}_2}\right]-2{\hat{c}}_3\right) \). This contrasts with the approach commonly followed, where the residuals are defined in the log-log model of the survival function, \( {\hat{u}}_i=\ln \left[1-{N}_i/N\right]-\hat{\ln}\left[1-{\hat{F}}_S\left({L}_i\right)\right] \).

  12. 12.

    The plot of Fig. 3 is referred to as the Zipf plot. See Stanley et al. (1995) for an early application to the distribution of firm size.

  13. 13.

    This result follows from the property that if g(x) and h(x) are power functions, then h(g(x)) is also a power function. In particular, the relationship between skills of the entrepreneur and profit-maximizing number of employees, L = h(e; w), will be a power function when the production technology is a linear homogeneous production function, for example a Cobb-Douglas or a CES.


  1. Aguinis, H., Gomez-Mejia, L., Martin, G., & Joo, H. (2018). CEO pay is indeed decoupled from CEO performance: charting a path for the future. Management Research: Journal of the Iberoamerican Academy of Management, 16(1), 117–136.

    Article  Google Scholar 

  2. Andriani, P., & McKelvey, B. (2009). From Gaussian to Paretian thinking: causes and implications of power laws in organizations. Organization Science, 20(6), 1053–1071.

    Article  Google Scholar 

  3. Axtell, R. (2001). Zipf distributions of U.S. firm sizes. Science, 293(5536), 1818–1820.

    Article  Google Scholar 

  4. Axtell, R. (2006). Firm sizes: Facts, formulae, fables and fantasies. Center on Social and Economic Dynamics Working Paper, No. 44.

  5. Bloom, N., & Van Reenen, J. (2007). Measuring and explaining management practices across firms and countries. Quarterly Journal of Economics, 122(4), 1351–1408.

    Article  Google Scholar 

  6. Broecke, S., Quintini, G., & Vandeweyer, M. (2017). Explaining international differences in wage inequality: Skills matter. Economics of Education Review, 60, 112–124.

    Article  Google Scholar 

  7. Clauset, A., Rohilla Shalizi, C., & Newman, M. E. J. (2009). Power-law distributions in empirical data. SIAM Review, 51(4), 661–703.

    Article  Google Scholar 

  8. Coase, R. H. (1937). The nature of the firm. Economica, 4, 386–405.

    Article  Google Scholar 

  9. Crawford, G. C. (2018). Skewed Opportunities: How the distribution of entrepreneurial inputs and outcomes Reconceptualizes a research domain. Proceedings-Academy of Management.

  10. Crawford, G. C., Aguinis, H., Lictenstein, B., Davidsson, P., & McKelvey, B. (2015). Power law distributions in entrepreneurship: implications for theory and research. Journal of Business Venturing, 30, 696–713.

    Article  Google Scholar 

  11. Davis, S., & Henrekson, M. (1999). Explaining national differences in the size and industry distribution of employment. Small Business Economics, 12(1), 59–83.

    Article  Google Scholar 

  12. Decker, R., Haltiwanger, J. R., Jarmin, R., & Miranda, J. (2018). Changing business dynamism and productivity: shocks vs. responsiveness. NBER Working Paper, 24236.

  13. Denrell, J., Fang, C., & Liu, C. (2014). Perspective - chance explanations in the management sciences. Organization Science, 26(3), 923–940.

    Article  Google Scholar 

  14. Ferrante, F. (2005). Revealing entrepreneurial talent. Small Business Economics, 25(2), 159–174.

    Article  Google Scholar 

  15. Fuchs-Schündeln, N., & Schündeln, M. (2005). Precautionary savings and self-selection: evidence from the German reunification ‘experiment’. The Quarterly Journal of Economics, 120(3), 1085–1120.

    Article  Google Scholar 

  16. Gabaix, X. (2016). Power laws in economics: an introduction. Journal of Economic Perspectives, 30(1), 185–206.

    Article  Google Scholar 

  17. Gabaix, X., & Landier, A. (2008). Why has CEO pay increased so much? Quarterly Journal of Economics, 123(1), 49–100.

    Article  Google Scholar 

  18. Geroski, P. A. (2000). The growth of firms in theory and practice. In N. Foss & V. Mahnke (Eds.), Chapter 8 of Competence, Governance and Entrepreneurship (pp. 168–186). Oxford, ISBN 10.0198297173/ISBN13:9780198297178: Oxford University Press.

    Google Scholar 

  19. Gibrat, R. (1931). Les Inégalités économiques. Paris: Recueil Sirey.

    Google Scholar 

  20. Green W. H. (2012). Econometric Analysis (7th Edition). London, England: Pearson Education Limited.

  21. Henrekson, M., & Johansson, D. (1999). Institutional effects on the evolution of the size distribution of firms. Small Business Economics, 12(1), 11–23.

    Article  Google Scholar 

  22. Hopenhayn, H. (1992). Entry, exit, and firm dynamics in long run equilibrium. Econometrica, 60(5), 1127–1150.

    Article  Google Scholar 

  23. Ijiri, Y., & Simon, H. A. (1967). A model of business firm growth. Econometrica, 35(2), 348–355.

    Article  Google Scholar 

  24. Joo, H., Aguinis, H., & Bradley, K. J. (2017). Not all nonnormal distributions are created equal: Improved theoretical and measurement precision. Journal of Applied Psychology, 102(7), 1022–1053.

    Article  Google Scholar 

  25. Jovanovic, B. (1982). Selection and the evolution of industry. Econometrica, 50(3), 649–670.

    Article  Google Scholar 

  26. Jovanovic, B. (1994). Firm formation with heterogeneous management and labor skills. Small Business Economics, 6(3), 185–191.

    Article  Google Scholar 

  27. Kihlstrom, R., & Laffont, J. J. (1979). A general equilibrium theory of firm formation based on risk aversion. Journal of Political Economy, 87(4), 719-748.

  28. Knudsen, T., Levinthal, D. A., & Winter, S. G. (2017). Systematic differences and random rates: reconciling Gibrat’s law with firm differences. Strategy Science, 2(2), 111–120.

    Article  Google Scholar 

  29. Lazear, E. P. (2004). Balanced skills and entrepreneurship. The American Economic Review, 94(2), 208–211.

    Article  Google Scholar 

  30. Lucas, R. (1978). On the size distribution of business firms. The Bell Journal of Economics, 9(2), 508–523

    Article  Google Scholar 

  31. McKelvey, B. (2004). Toward a complexity science of entrepreneurship. Journal of Business Venturing, 19, 313–341.

    Article  Google Scholar 

  32. Medrano-Adan, L., Salas-Fumás, V., & Sanchez-Asin, J. J. (2015). Heterogeneous entrepreneurs from occupational choices in economies with minimum wage. Small Business Economics, 44, 597–619.

    Article  Google Scholar 

  33. Medrano-Adán, L., Salas-Fumás, V., & Sanchez-Asin, J. J. (2019). Firm size and productivity from occupational choices. Small Business Economics, 53(1), 243–267.

    Article  Google Scholar 

  34. Melitz, M. (2003). The impact of trade on intra-industry reallocations and aggregate industry productivity. Econometrica, 71(6), 1695–1725.

    Article  Google Scholar 

  35. Moral Benito, E. (2018). Growing by learning: firm-level evidence on the size-productivity nexus. SERIEs, 9(1), 65–90.

    Article  Google Scholar 

  36. Murphy, K., Shleifer, A., & Vishny, R. (1991). The allocation of talent: implications for growth. The Quarterly Journal of Economics, 106(2), 503-530.

  37. Ortín-Ángel, P., & Vendrell-Herrero, F. (2010). University spin-offs vs. other NTBFs: productivity differences at the outset and evolution. In Searle Center on Law, Regulation, and Economic Growth. Working Papers, 2010–027. Chicago: Northwestern University.

    Google Scholar 

  38. Penrose, E. (1959). The theory of the growth of the firm. New York: John Wiley.

    Google Scholar 

  39. Ramsey, J. B. (1969). Tests for specification errors in classical linear least-squares regression analysis. Journal of the Royal Statistical Society. Series B (Methodological), 31(2), 350–371

    Article  Google Scholar 

  40. Roper, S. (1998). Entrepreneurial characteristics, strategic choice and small business performance. Small Business Economics, 11, 11–24.

    Article  Google Scholar 

  41. Rosen, S. (1981). The economics of superstars. The American Economic Review, 71(5), 845–858

    Google Scholar 

  42. Rosen, S. (1982). Authority, control, and the distribution of earnings. The Bell Journal of Economics, 13(2), 311–323.

    Article  Google Scholar 

  43. Schumpeter, J. A. (1934). The theory of economic development: an inquiry into profits, capital, credit, interest and the business cycle. Cambridge, MA: Harvard University Press.

  44. Shepherd, D. A., Wennberg, K., Suddaby, R., & Wiklund, J. (2019). What are we explaining? A review and agenda on initiating, engaging, performing, and contextualizing entrepreneurship. Journal of Management, 45(1), 159–196.

    Article  Google Scholar 

  45. Simon, H. A. (1955). On a class of skew distribution functions. Biometrika, 42(3/4), 425–440.

    Article  Google Scholar 

  46. Stanley, M. H. R., Buldyrev, S. V., Havlin, S., Mategna, R. N., Salinger, M. A., & Stanley, H. E. (1995). Zipf plots and the size distribution of firms. Economic Letters, 49(4), 453–457.

    Article  Google Scholar 

  47. Storey, D. (1994). Understanding the small business sector. London: Routledge.

    Google Scholar 

  48. Vendrell-Herrero, F., González-Pernía, J. L., & Peña-Legazkue, I. (2014). Do incentives matter to promote high technology-driven entrepreneurial activity? International Entrepreneurship and Management Journal, 10(1), 43–66.

    Article  Google Scholar 

  49. Wooldridge, J. M. (2019). Introductory econometrics: a modern approach (7th Edition). Cincinati, O: South Western Cengage Learning.

Download references


This study was financially supported by the Spanish Ministry of Economy and Competitiveness-FEDER (ECO2017-86305-C4-3-R) and by the Government of Aragón (Group Reference CREVALOR: S42_17R) and co-financed with FEDER 2014-2020 “Construyendo Europa desde Aragón.”

Author information



Corresponding author

Correspondence to Vicente Salas-Fumás.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information


(PDF 465 kb)


Appendix 1. Proofs of the claims

Lemma 1.

If x is a random variable with support [xm, +∞), then there exists an x* > 0 such that the probability density function (PDF) is strictly decreasing and convex for all x > x*.

Proof. The demonstration is made by reductio ad absurdum (reduction to absurdity). Consider a random x variable with support [xm, +∞) (xm may be any real number or −∞).

The probability density function satisfies that \( \underset{-\infty }{\overset{+\infty }{\int }}f(x) dx=1,f(x)\ge 0\kern0.5em \forall x \). If f(x) is not decreasing in the right tail (and the support is [xm, +∞)), then there exist δ > 0 and x* > 0, such that f(x) > δ for all x > x*, but then \( \underset{x^{\ast }}{\overset{+\infty }{\int }}f(x) dx>\underset{x^{\ast }}{\overset{+\infty }{\int }}\delta dx=+\infty \), which is absurd. Consequently, f(x) is decreasing in the right tail; more precisely, there exists x* such that f(x) is strictly decreasing for all x > x*. Moreover, if f(x) is concave for all x > x*, then there is a value x** such that f(x**) = 0, which is absurd, since we are assuming that f(x) > 0 for all x ∈ [xm, +∞). Therefore, f(x) must be strictly decreasing and convex in the right tail. ■.

Proof of claim 1

In equilibrium, the individuals who become entrepreneurs are those with skill levels higher or equal to e*, so that the distribution of skills in the group of entrepreneurs is just the left-truncated distribution of skills in the population. Therefore, its support is [e, eM] for eM finite, and [e, +∞) otherwise; and its probability cumulative function and density probability function are characterized by.

\( {F}_{ent\; skills}(e)=\frac{F(e)-F\left({e}^{\ast}\right)}{F\left({e}_M\right)-F\left({e}^{\ast}\right)},\kern0.75em {f}_{ent\; skills}(e)=\frac{f(e)}{F\left({e}_M\right)-F\left({e}^{\ast}\right)}\kern1em \left(\forall e\ge {e}^{\ast}\right) \) (*).

since, for any continuous random variable X with support [xm, xM] and probability density function g(x) and cumulative distribution function G(x), the truncated distribution of X for the values in [a, b] (xm ≤ a < b ≤ xM) is characterized by \( {g}_{truncated}(x)=\frac{g(x)}{G(b)-G(a)} \) and \( {G}_{truncated}(x)=\frac{G(x)-G(a)}{G(b)-G(a)} \) (for any x such that a ≤ x ≤ b).

On the other hand, Firm size S, measured by number of employees, is a function of the entrepreneur’s skills e, S = h(e; w), for e ≥ e*; so that the distribution of firm size is just the transformation (by “h”) of the distribution of the entrepreneurs’ skills. And, for any continuous random variable X with probability density function fX(x) and cumulative distribution function FX(x), and Y = h(X), where h is invertible (and monotonous and differentiable), the probability cumulative function and the density probability function of Y are characterized by FY(y) = FX(h−1(y)) and \( {f}_Y(y)={f}_X\left({h}^{-1}(y)\right)\frac{d\kern0.1em {h}^{-1}(y)}{dy} \). By applying these expressions to the distribution of entrepreneurs’ skills, expression (*) above, and S = h(e; w), we directly obtain that the probability cumulative function and PDF of S are characterized by

$$ {F}_S(s)=\frac{F\left({h}^{-1}(s)\right)-F\left({e}^{\ast}\right)}{F\left({e}_M\right)-F\left({e}^{\ast}\right)},{f}_S(s)=\frac{f\left({h}^{-1}(s)\right)}{F\left({e}_M\right)-F\left({e}^{\ast}\right)}\frac{\mathrm{d}\kern0.28em {h}^{-1}(s)}{\mathrm{d}\kern0.28em s}\kern1em \forall s\ge {S}_{\mathrm{min}} $$

Where Smin = h(e; w). ■.

Proof of claim 2.

On the one hand, for reasonable production functions, the (individual) labor demand will be increasing and convex in the entrepreneur’s skill, \( L=h\left(e;w\right),\frac{\partial h}{\partial e}>0,\frac{\partial^2h}{\partial {e}^2}>0 \), so that its inverse, h−1(), will be strictly increasing and concave, \( \frac{dh^{-1}(s)}{ds}>0 \) and \( \frac{d^2{h}^{-1}(s)}{ds^2}<0 \). On the other hand, for reasonable distributions of skills, with support [em, +∞), there exists e1 such that f ' (x) < 0 and f ''(x) > 0, for x ≥ e1 ≥ em. So that, by assumption,

\( \frac{d{h}^{-1}(s)}{ds}>0,\frac{d^2{h}^{-1}(s)}{d{s}^2}<0,f\hbox{'}(x)<0,f\hbox{'}\hbox{'}(x)>0. \) (*)

From [2], the probability density function of the distribution of firm size is \( {f}_S(s)=\frac{f\left({h}^{-1}(s)\right)}{F\left({e}_M\right)-F\left({e}^{\ast}\right)}\frac{dh^{-1}(s)}{ds} \) and its first derivative is

$$ \frac{\mathrm{d}{f}_S(s)}{\mathrm{d}s}=\frac{1}{F\left({e}_M\right)-F\left({e}^{\ast}\right)}\left[f^{\prime}\left({L}^{-1}(s)\right)\frac{d{L}^{-1}(s)}{ds}\frac{d{L}^{-1}(s)}{ds}+f\left({L}^{-1}(s)\right)\frac{d^2{L}^{-1}(s)}{d{s}^2}\right] $$

It directly follows, from the assumptions (*), that f ' (h−1(s)) < 0, \( \frac{dh^{-1}(s)}{ds} \) > 0, f(h−1(s))>0 \( \frac{d^2{h}^{-1}(s)}{ds^2} \) < 0, and then both summands in the above expression of \( \frac{\mathrm{d}{f}_S(s)}{\mathrm{d}s} \) are negative; i.e. fS(s) is strictly decreasing.

On the other hand, the second derivative is given by

$$ {\displaystyle \begin{array}{l}{\beta \theta}^{\frac{1}{\beta }}{\left(\frac{1-\beta }{w^{\ast }}\right)}^{\frac{\left(1-\beta \right)}{\beta }}{\left({e}^{\ast}\right)}^{\frac{\left(\beta +\tau \right)}{\beta }={w}^{\ast }},\\ {}\underset{e_m}{\overset{e^{\ast }}{\int }} dF(e)={\left(\frac{\theta \left(1-\beta \right)}{w^{\ast }}\right)}^{\frac{1}{\beta }}\underset{e^{\ast }}{\overset{e_m}{\int }}{e}^{\left(\beta +\tau \right)/\beta } dF(e)\end{array}} $$
$$ {\displaystyle \begin{array}{c}\frac{{\mathrm{d}}^2{f}_S(s)}{\mathrm{d}{s}^2}=\frac{1}{F\left({e}_M\right)-F\left({e}^{\ast}\right)}\left[f"\left({h}^{-1}(s)\right){\left(\frac{d{h}^{-1}(s)}{ds}\right)}^3+\right.\\ {}\kern1.75em +2f^{\prime}\left({h}^{-1}(s)\right)\left(\frac{d{h}^{-1}(s)}{ds}\right)\frac{d^2{h}^{-1}(s)}{d{s}^2}+\\ {}\begin{array}{l}+f^{\prime}\left({h}^{-1}(s)\right)\frac{d{h}^{-1}(s)}{ds}\frac{d^2{h}^{-1}(s)}{d{s}^2}+\\ {}\left.+f\left({h}^{-1}(s)\right)\frac{d^3{h}^{-1}(s)}{d{s}^3}\right]\end{array}\end{array}} $$

From the assumptions, expression (*) above in this proof, it is obvious that the three first summands in the above expression are positive. The fourth one is also positive for labor demands of power function (or polynomial) type. Consequently, fS''(s) > 0. ■

Proof of claim 3.

If the individual labor demand is a power function, \( h\left(e;w\right)={a}_1\kern0.5em {e}^{b_1} \), then 

\( {h}^{-1}(s)={\left(s/{a}_1\right)}^{1/{b}_1} \) and \( \frac{dh^{-1}(s)}{ds}=\frac{1}{b_1{a}_1}{\left(\frac{s}{a_1}\right)}^{\left(1-{b}_1\right)/{b}_1} \).

Sufficient condition: (if f(x) is a power law, then fS(x) is also a power law).

On the one hand, if the distribution of skills follows a power law, then its probability density function may be written asf(x) = C x−1 − α (where eM =  +  ∞ , C = αemα, F(eM) − F(e) = emα(e)α and F(e) =  − emα(e)α), and from [3], the probability density function of the distribution of size is given by

$$ {f}_S(s)=\frac{f\left({h}^{-1}(s)\right)}{F\left({e}_M\right)-F\left({e}^{\ast}\right)}\frac{d{h}^{-1}(s)}{ds}=\frac{C\;{\left({\left(s/{a}_1\right)}^{1/{b}_1}\right)}^{-1-\alpha }}{\left[F\left({e}_M\right)-F\left({e}^{\ast}\right)\right]{b}_1{a}_1}{\left(\frac{s}{a_1}\right)}^{\left(1-{b}_1\right)/{b}_1}=C^{\prime }{\left(\frac{s}{a_1}\right)}^{-1-\left(\alpha /{b}_1\right)} $$

which is a power law as well, but with constant parameter \( C\hbox{'}=\frac{C\;}{\left[F\left({e}_M\right)-F\left({e}^{\ast}\right)\right]{b}_1{a}_1} \) and power parameter (α/b1).

Necessary condition: (if fS(x) is a power law, then f(x) must be a power law as well).

First, remember that the DFS follows a power law if and only if the elasticity of its probability density function is constant (and let us denote this elasticity κS):

$$ {\eta}_S=\frac{s}{f_S(s)}\frac{\mathrm{d}{f}_S(s)}{\mathrm{d}s}={\kappa}_S $$

Let us assume that this elasticity is constant and equal to κS. From [3], \( {f}_S(s)=\frac{f\left({h}^{-1}(s)\right)}{F\left({e}_M\right)-F\left({e}^{\ast}\right)}\frac{dh^{-1}(s)}{ds} \), so that the above elasticity is given by

$$ {\eta}_S=\frac{s}{f_S(s)}\frac{\mathrm{d}{f}_S(s)}{\mathrm{d}s}=\frac{s\frac{1}{F\left({e}_M\right)-F\left({e}^{\ast}\right)}\left[f^{\prime}\left({h}^{-1}(s)\right)\frac{d{h}^{-1}(s)}{ds}\frac{d{h}^{-1}(s)}{ds}+f\left({h}^{-1}(s)\right)\frac{d^2{h}^{-1}(s)}{d{s}^2}\right]}{\frac{f\left({h}^{-1}(s)\right)}{F\left({e}_M\right)-F\left({e}^{\ast}\right)}\frac{d{h}^{-1}(s)}{ds}}={\eta}_S=\frac{s\;f^{\prime}\left({h}^{-1}(s)\right)}{f\left({h}^{-1}(s)\right)}\frac{d{h}^{-1}(s)}{ds}+\frac{s\frac{d^2{h}^{-1}(s)}{d{s}^2}}{\frac{d{h}^{-1}(s)}{ds}} $$

Remember that, if labor demand is a power function, \( h\left(e;w\right)={a}_1\kern0.5em {e}^{b_1} \), then \( {h}^{-1}(s)={\left(s/{a}_1\right)}^{1/{b}_1} \), \( {h}^{-1}(s)={\left(\frac{s}{a_1}\right)}^{1/{b}_1} \), \( \frac{dh^{-1}(s)}{ds}=\frac{1}{b_1{a_1}^{1/{b}_1}}{s}^{\left(1/{b}_1\right)-1} \) and \( \frac{d^2{h}^{-1}(s)}{ds^2}=\frac{\left(1-{b}_1\right)/{b}_1}{b_1{a_1}^{1/{b}_1}}{s}^{\left(1/{b}_1\right)-2} \). Note also that \( {h}^{-1}(s)={b}_1s\frac{dh^{-1}(s)}{ds} \) or equivalently, \( \frac{dh^{-1}(s)}{ds}=\frac{1}{b_1{a_1}^{1/{b}_1}}{s}^{\left(1/{b}_1\right)-1}=\frac{1}{b_1s}{\left(\frac{s}{a_1}\right)}^{\left(1/{b}_1\right)}=\frac{1}{b_1s}{h}^{-1}(s) \).

By substituting \( \frac{dh^{-1}(s)}{ds}=\frac{1}{b_1{a_1}^{1/{b}_1}}{s}^{\left(1/{b}_1\right)-1} \), \( \frac{d^2{h}^{-1}(s)}{ds^2}=\frac{\left(1-{b}_1\right)/{b}_1}{b_1{a_1}^{1/{b}_1}}{s}^{\left(1/{b}_1\right)-2} \), and \( {h}^{-1}(s)={b}_1s\frac{dh^{-1}(s)}{ds} \), into the last expression of the last elasticity of ηS we get

$$ {\eta}_S=\frac{s\;f^{\prime}\left({h}^{-1}(s)\right)}{f\left({h}^{-1}(s)\right)}\frac{h^{-1}(s)}{b_1\;s}+\frac{s\frac{\left(1-{b}_1\right)/{b}_1}{{b_1{a}_1}^{1/{b}_1}}{s}^{\left(1/{b}_1\right)-2}}{\frac{1}{{b_1{a}_1}^{1/{b}_1}}{s}^{\left(1/{b}_1\right)-1}}=\frac{1}{b_1}\frac{\;{h}^{-1}(s)}{f\left({h}^{-1}(s)\right)}f^{\prime}\left({h}^{-1}(s)\right)+\frac{1-{b}_1}{b_1} $$

Let us now make the change of variable x = h−1(s). Then,

$$ {\eta}_S=\frac{1}{b_1}\frac{x}{f(x)}f\hbox{'}(x)+\frac{1-{b}_1}{b_1} $$

The DFS follows a power law if this elasticity is constant,

$$ {\displaystyle \begin{array}{c}{\eta}_S=\frac{1}{b_1}\frac{x}{f(x)}f\hbox{'}(x)+\frac{1-{b}_1}{b_1}={\kappa}_S\iff \\ {}\frac{x}{f(x)}f\hbox{'}(x)={b}_1\left({\kappa}_S+1\right)-1\end{array}} $$

This is a simple differential equation, and the “unique” solution is \( f(x)={C}_1{x}^{b_1\left({\kappa}_S+1\right)-1} \), which is the density probability function of a Pareto distribution (for an appropriate value of the constant C1).

Alternatively, we can prove that f(x) must be a power law as follows. If labor demand is a power function and the distribution of size follows a power law, fS(s) = C1s−1 − k, then, from [3]

$$ {f}_S(s)={C}_1\kern0.28em {s}^{-1-k},\kern.5em \mathrm{and}{f}_S(s)=\frac{f\left({h}^{-1}(s)\right)}{F\left({e}_M\right)-F\left({e}^{\ast}\right)}\frac{d{h}^{-1}(s)}{ds}\Rightarrow {C}_1\kern0.28em {s}^{-1-k}=\frac{1}{\left[F\left({e}_M\right)-F\left({e}^{\ast}\right)\right]{b}_1{a}_1}{\left(\frac{s}{a_1}\right)}^{\left(1-{b}_1\right)/{b}_1}f\left({\left(s/{a}_1\right)}^{1/{b}_1}\right) $$

After rearranging some terms, we get

$$ f\left({\left(s/{a}_1\right)}^{1/{b}_1}\right)={C}_1\left[F\left({e}_M\right)-F\left({e}^{\ast}\right)\right]{b}_1{a_1}^{-k}{\left({\left(s/{a}_1\right)}^{1/{b}_1}\right)}^{\left(-1-{kb}_1\right)} $$

So that the probability density function of the distribution of skills may be written as (after changing variables, \( x={\left(s/{a}_1\right)}^{1/{b}_1} \))

$$ f(x)=C\;{x}^{\left(-1-{kb}_1\right)} $$

which is a power law as well (but with power parameter k b1 and constant/scale parameter C = C1[F(eM) − F(e)]b1a11 − k. ■.

Appendix 2. A closed equilibrium solution from occupational choices and power law distribution of entrepreneurial skills

The Pareto distribution of skills has the advantage that it allows for closed solutions to the occupational choice equilibrium, which facilitates the comparative static analysis of changes in the DFS to changes in the parameters of the model. In this appendix, we present the market equilibrium solution from occupational choices for the simple production function Q = θ e(1 + β)L1 − β and a distribution of entrepreneurial skill in the working population that follows a Pareto distribution (with minimum value em and power parameter α). It can be shown that equilibrium exists and is unique for values of the parameters that satisfy the conditionFootnote 13 (α − 1)β > 1. The level of skill for which the individual is indifferent between working as employee or working as an entrepreneur and the equilibrium salary are given by (see Medrano-Adán et al. (2019) for the details):

$$ {e}^{\ast }={e}_m{\left(\frac{\alpha -\beta -1}{\left(\alpha -1\right)\beta -1}\right)}^{1/\alpha }{w}^{\ast }=\theta {\left(1-\beta \right)}^{1-\beta }{\beta}^{\beta }{e_m}^{\beta +1}{\left(\frac{\alpha -\beta -1}{\left(\alpha -1\right)\beta -1}\right)}^{\left(\beta +1\right)/\alpha } $$

The distribution of firm size is (also) a Pareto distribution with power parameter \( \frac{\alpha\;\beta }{\left(\beta +1\right)} \) and minimum size value \( {S}_{\mathrm{min}}=h\left({e}^{\ast };{w}^{\ast}\right)=a\;{\left({e}^{\ast}\right)}^b=\frac{\left(1-\beta \right)}{\beta } \). The probability density function and cumulative distribution function are given by (for x ≥ Smin = (1 − β)/β)

$$ {f}_S(x)=\frac{\alpha\;\beta }{\left(\beta +1\right)}{S}_{\mathrm{min}}^{\frac{\alpha\;\beta }{\left(\beta +1\right)}}{x}^{-1-\frac{\alpha\;\beta }{\left(\beta +1\right)}},\kern2.00em {F}_S(x)=1-{\left(\frac{S_{\mathrm{min}}}{x}\right)}^{\frac{\alpha\;\beta }{\left(\beta +1\right)}} $$

The density and distribution functions only depend on the power parameter of the distribution of skills α, and on the parameter of the organization technology, β. We can easily prove the following analytical results:

  1. 1.

    The average firm size, given by \( \overline{S}=\frac{\alpha \left(1-\beta \right)}{\left(\alpha -1\right)\beta -1} \), is strictly decreasing in α, the concentration of the distribution of skills, and in the organizational size diseconomies parameter, β.

  2. 2.

    The minimum firm size is strictly decreasing in β,

  3. 3.

    The power parameter of the distribution function of firm size \( \frac{\alpha\;\beta }{\left(\beta +1\right)} \) is strictly increasing in both α and β.

  4. 4.

    The quantiles of the firm size distribution \( Quantile(q)=\frac{\left(1-\beta \right)}{\beta }{\left(1-q\right)}^{-\frac{1+\beta }{\alpha \beta}} \) are strictly decreasing in both α and β).

The average size of firms in the occupational choice equilibrium increases with lower organizational size diseconomies, i.e., with lower intensity of time of the entrepreneur in the supervision of the employees under direction, lower β, and with higher dispersion of skills in the population, lower α. Lower organizational size diseconomies, lower β, also implies larger minimum firm size, i.e., lower number of micro firms in the market equilibrium. The proportion of employees in larger firms, for example in the top 1% quantile of the size distribution, decreases with α and with β.

The survival function of the DFS, the complementary of the cumulative distribution function, is a power function:

$$ S{F}_S(x)=1-{F}_s(x)={\left(\frac{S_{\mathrm{min}}}{x}\right)}^{\frac{\alpha\;\beta }{\left(\beta +\tau \right)}}\kern1em \left(\forall x\ge {S}_{\mathrm{min}}=\left(1-\beta \right)/\beta \right) $$

So that its plot in logarithmic scale (the so-called Zipf plot) is linear; there is a negative linear relationship between the log of the proportion of firms with L or more employees and the number of employees L.

$$ \ln \left[1-{F}_S(x)\right]=\frac{\alpha\;\beta }{\left(\beta +\tau \right)}\left[\ln \left({S}_{\mathrm{min}}\right)-\ln (x)\right] $$

The DFS satisfies the property of power law distributions of constant elasticity, equal to \( \frac{\alpha\;\beta }{\left(\beta +1\right)} \), between the proportion of firms above a value of the variable L and the value of L. In the Zipf distribution the power parameter would be equal to 1, \( \frac{\alpha\;\beta }{\left(\beta +1\right)} \)= 1, which implies that (α − 1)β = 1. For values of the parameters satisfying this condition, the occupational choice equilibrium would not have a finite equilibrium (average size of firms tends to infinity, for example). In other words, for economically meaningful occupational choices equilibrium, the parameters of the model must satisfy (α − 1)β > 1 and the power parameter must be greater than one. Axtell (2001) estimated a power parameter \( \frac{\alpha\;\beta }{\left(\beta +1\right)} \)= 1.059, greater than one, although, as we show in the main text, we reject the null hypothesis that the US distribution of firm size really follows a power law.

Appendix 3. Test of the power law with Spanish data on firm size

For robustness purpose we present empirical results additional to those in section 4 on the power law distribution of firm size, now with Spanish data. The Spanish official statistical office, INE, publishes annual binned data from 1999 till 2019 on the distribution of firm size (see, with number of firms in each of the 11 size classes with limits {1, 2, 5, 9, 19, 49, 99, 199, 499, 999, 4999, more than 5000}. For each of the years we estimate Eqs. [5] and [7] in the main text and perform the Ramsey RESET and omitted variable specification tests.

For each of the 21 years considered (1999–2019), the estimated value of δ is in the interval (− 0.0133, − 0.0328), and the p value of the test of the mull hypothesis that δ = 0 is always lower than 0.00604 (with mean value equal to 0.00070). The Ramsey RESET test gives similar results, with p values lower than 0.0051 (with mean value equal to 0.0006). Table 5 shows a summary of the estimation results: the hypothesis of δ = 0 is rejected at a high level of significance (higher than 99%) for all the years.

Consequently, the specification tests do not support the conclusion that the data on the distribution of firm size in Spain follows a power law, similar to the result in the main text for US firm size data.

Table 5 Estimations of the distributions of firm size with Spanish data: log-linear and log-quadratic model: selected years in the period 1999 to 2019

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Medrano-Adán, L., Salas-Fumás, V. The added value of management skill in the explanation of the distribution of firm size. Small Bus Econ (2021).

Download citation


  • Occupational choice
  • Entrepreneurial skill
  • Distribution of firm size
  • Power law

JEL Classifications

  • J24
  • K31
  • L11
  • L25
  • D01
  • D24
  • D31
  • L26