Abstract
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.
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Notes
 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.
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.
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.
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 cutoff” 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 heavytailed 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.
See also McKelvey (2004) on complexity science and entrepreneurship.
 6.
 7.
As complement, Appendix 2 shows the closed solution to the market equilibrium from occupational choices when the distribution of skill is Pareto.
 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 noneconomic 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.
The Ramsey Regression Equation Specification Error Test (RESET), Ramsey (1969), is a general specification test, which tests whether nonlinear combinations of the fitted values help explain the dependent variable. If the null hypothesis that all coefficients of nonlinear combinations of the fitted values are zero is rejected, then the model suffers from misspecification (Wooldridge, 2019; Green, 2012).
 10.
With binned data, the likelihood function can be approximated by \( \overline{L}\left(\right)\prod \limits_{i=1}^{13}{\left[F\left({L}_i\right)F\left({L}_{i1}\right)\right]}^{n_i} \). 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 loglikelihood function to maximize is \( Log\left[\overset{\frown }{L}\right]=N\ln \left(\frac{1}{\zeta \left(1+\rho \right)}\right)+\sum \limits_{i=1}^{13}{n}_i\ln \left[\sum \limits_{k=1}^{Li}{k}^{1\rho }\sum \limits_{k=1}^{Li1}{k}^{1\rho}\right] \) and the optimal solution is \( {\hat{\rho}}_{ML}=0.501547 \).
 11.
We minimize the sum of squares of residuals in the “original” variable, number of firms,\( {\hat{u}}_i=\left({N}_i/N\right){\hat{F}}_s\left({L}_i\right)=\frac{N_i}{N}\frac{1}{22{\hat{c}}_3}\left( Erfc\left[\frac{\hat{c}\ln \left({L}_i\right)}{{\hat{c}}_2}\right]2{\hat{c}}_3\right) \). This contrasts with the approach commonly followed, where the residuals are defined in the loglog 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.
 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 profitmaximizing 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 CobbDouglas or a CES.
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Funding
This study was financially supported by the Spanish Ministry of Economy and CompetitivenessFEDER (ECO201786305C43R) and by the Government of Aragón (Group Reference CREVALOR: S42_17R) and cofinanced with FEDER 20142020 “Construyendo Europa desde Aragón.”
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Appendices
Appendix 1. Proofs of the claims
Lemma 1.
If x is a random variable with support [x_{m}, +∞), 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 [x_{m}, +∞) (x_{m} 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 [x_{m}, +∞)), 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 ∈ [x_{m}, +∞). 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 lefttruncated distribution of skills in the population. Therefore, its support is [e^{∗}, e_{M}] for e_{M} 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 [x_{m}, x_{M}] and probability density function g(x) and cumulative distribution function G(x), the truncated distribution of X for the values in [a, b] (x_{m} ≤ a < b ≤ x_{M}) 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 f_{X}(x) and cumulative distribution function F_{X}(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 F_{Y}(y) = F_{X}(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
Where S_{min} = 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 [e_{m}, +∞), there exists e_{1} such that f ' (x) < 0 and f ''(x) > 0, for x ≥ e_{1} ≥ e_{m}. 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
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. f_{S}(s) is strictly decreasing.
On the other hand, the second derivative is given by
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, f_{S}''(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 f_{S}(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 e_{M} = + ∞ , C = αe_{m}^{α}, F(e_{M}) − F(e^{∗}) = e_{m}^{α}(e^{∗})^{−α} and F(e^{∗}) = − e_{m}^{α}(e^{∗})^{−α}), and from [3], the probability density function of the distribution of size is given by
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/b_{1}).
Necessary condition: (if f_{S}(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 κ_{S}):
Let us assume that this elasticity is constant and equal to κ_{S}. 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
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 η_{S} we get
Let us now make the change of variable x = h^{−1}(s). Then,
The DFS follows a power law if this elasticity is constant,
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 C_{1}).
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, f_{S}(s) = C_{1} s^{−1 − k}, then, from [3]
After rearranging some terms, we get
So that the probability density function of the distribution of skills may be written as (after changing variables, \( x={\left(s/{a}_1\right)}^{1/{b}_1} \))
which is a power law as well (but with power parameter k b_{1} and constant/scale parameter C = C_{1}[F(e_{M}) − F(e^{∗})]b_{1}a_{1}^{1 − 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)}L^{1 − β} and a distribution of entrepreneurial skill in the working population that follows a Pareto distribution (with minimum value e_{m} and power parameter α). It can be shown that equilibrium exists and is unique for values of the parameters that satisfy the condition^{Footnote 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 MedranoAdán et al. (2019) for the details):
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 ≥ S_{min} = (1 − β)/β)
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.
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.
The minimum firm size is strictly decreasing in β,

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.
The quantiles of the firm size distribution \( Quantile(q)=\frac{\left(1\beta \right)}{\beta }{\left(1q\right)}^{\frac{1+\beta }{\alpha \beta}} \) 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:
So that its plot in logarithmic scale (the socalled 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.
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 https://www.ine.es/jaxiT3/Tabla.htm?t=299&L=0), 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.
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MedranoAdán, L., SalasFumás, V. The added value of management skill in the explanation of the distribution of firm size. Small Bus Econ (2021). https://doi.org/10.1007/s1118702100447y
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Keywords
 Occupational choice
 Entrepreneurial skill
 Distribution of firm size
 Power law
JEL Classifications
 J24
 K31
 L11
 L25
 D01
 D24
 D31
 L26