We analyze a database comprising quarterly sales of 55624 pharmaceutical
products commercialized by 3939 pharmaceutical firms in the period 1992--2001.
We study the probability density function (PDF) of growth in firms and product
sales and find that the width of the PDF of growth decays with the sales as a
power law with exponent $\beta = 0.20 \pm 0.01$. We also find that the average
sales of products scales with the firm sales as a power law with exponent
$\alpha = 0.57 \pm 0.02$. And that the average number products of a firm scales
with the firm sales as a power law with exponent $\gamma = 0.42 \pm 0.02$. We
compare these findings with the predictions of models proposed till date on
growth of business firms

Amaral L. A. N.

Buldyrev S. V.

Coase R. H.

Dongfeng Fu

F Pammolli

H. E Stanley

Ijiri Y.

K Matia

M Riccaboni

S. V Buldyrev

Wyart M.

English

arXiv

Riccaboni, M

Archivio della ricerca della Scuola IMT Alti Studi Lucca

Statistical properties of business firms structure and growth RID D-4102-2009

Crossref

Statistical properties of business firms structure and growth

Pammolli, Fabio

Florence Research

EUROPHYSICSLETTERSPublished under the scientific responsibility of theEUROPEAN PHYSICAL SOCIETYIncorporatingJOURNAL DE PHYSIQUE LETTRES • LETTERE AL NUOVO CIMENTOOFFPRINTVol. 67 • Number 3 • pp. 498–503Statistical properties of business firmsstructure and growth∗ ∗ ∗K. Matia, Dongfeng Fu, S. V. Buldyrev, F. Pammolli,M. Riccaboni and H. E. Stanleyor directly to EDP Sciences:17 av. du Hoggar   B.P. 112   91944 Les Ulis Cedex A FranceTel. 33 (0)1 69 18 75 75   Fax 33 (0)1 69 86 06 78   subscribers@edpsciences.orgEditor-in-ChiefProf. Denis JéromeLab. Physique des Solides - Université Paris-Sud91405 Orsay - Francejerome@lps.u-psud.fr Order through your subscription agencySubscription 200424 issues - Vol. 65-68 (6 issues per vol.)ISSN: 0295-5075 - ISSN electronic: 1286-4854France & EU  1 678 €(VAT included)Rest of the World  1 678 €(without VAT)Europhysics Letter was launched more than fifteen years ago by the European Physical Society, the SociétéFrançaise de Physique, the Società Italiana di Fisica and the Institute of Physics (UK) and owned now by17 National Physical Societies/Institutes. Europhysics Letters aims to publish short papers containing non-trivial new results, ideas, concepts,experimental methods, theoretical treatments, etc. which are of broad interest and importance to one orseveral sections of the physics community.Europhysics letters provides a platform for scientists from all over the world to offer their results to aninternational readership.Payment:❐ Check enclosed payable to EDP Sciences❐ Please send me a pro forma invoice❐ Credit card:❐ Visa ❐ Eurocard ❐ American ExpressValid until:Card No: Signature:❐ Please send me a free sample copyInstitution/Library:  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Name:  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Position:  . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Address:  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ZIP-Code:  . . . . . . . . . . . . . . . . . . . . . . . . . . .City:  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Country:  . . . . . . . . . . . . . . . . . . . . . . . . . . . . .E-mail:  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .EUROPHYSICSLETTERSTaking full advantage of the service on Internet,please choose the fastest connection:http://www.edpsciences.orghttp://edpsciences.nao.ac.jphttp://edpsciences-usa.orghttp://www.epletters.chStaff Editor: Yoanne SobieskiEurophysics Letters, European Physical Society, 34 rue M. Seguin, 68060 Mulhouse Cedex, FranceEditorial Director: Angela Oleandri Director of publication: Jean-Marc QuilbéProduction Editor: Paola MarangonPublishers: EDP Sciences S.A., France - Società Italiana di Fisica, ItalyEurophys. Lett., 67 (3), pp. 498–503 (2004)DOI: 10.1209/epl/i2004-10070-4EUROPHYSICS LETTERS 1 August 2004Statistical properties of business firmsstructure and growthK. Matia1, Dongfeng Fu1, S. V. Buldyrev1, F. Pammolli2,M. Riccaboni2 and H. E. Stanley11 Center for Polymer Studies and Department of Physics, Boston UniversityBoston, MA 02215 USA2 Facolta` di Economia, Universita` di Firenze and CERMVia Banchi di Sotto 55, 53100 Siena Italy(received 10 March 2004; accepted in final form 17 May 2004)PACS. 89.90.+n – Other topics in areas of applied and interdisciplinary physics.PACS. 05.45.Tp – Time series analysis.PACS. 05.40.Fb – Random walks and Levy flights.Abstract. – We analyze a database comprising quarterly sales of 55624 pharmaceuticalproducts commercialized by 3939 pharmaceutical firms in the period 1992–2001. We study theprobability density function (PDF) of growth in firms and product sales and find that the widthof the PDF of growth decays with the sales as a power law with exponent β = 0.20 ± 0.01.We also find that the average sales of products scales with the firm sales as a power law withexponent α = 0.57 ± 0.02. And that the average number products of a firm scales with thefirm sales as a power law with exponent γ = 0.42± 0.02. We compare these findings with thepredictions of models proposed till date on growth of business firms.In economics there are unsolved problems that involve interactions among a large numberof subunits [1–3]. One of these problems is the structure of a business firm and its growth [2,4].As in many physical models, decomposition of a firm into its constituent parts is an appropriatestarting place for constructing a model. Indeed, the total sales of a firm is comprised of a largenumber of product sales. Previously accurate data on the “microscopic” product sales havebeen unavailable, and hence it has been impossible to test the predictions of various models.Here we analyze a new database, the Pharmaceutical Industry Database (PHID) which recordsquarterly sales figures of 55624 pharmaceutical products commercialized by 3939 firms in theEuropean Union and North America from September 1991 to June 2001. We shall see thatthese data support the predictions of a simple model, and at the same time the data do notsupport the microscopic assumptions of that model. In this sense, the model has the samestatus as many statistical physics models, in that the predictions can be in accord with dataeven though the details of microscopic interactions are not. The assumptions of this simplemodel given in ref. [5] are as follows: i) Firms tends to organize themselves into multipledivisions once they attain a particular size. ii) The minimum size of firms in a particulareconomy comes from a broad distribution. iii) Growth rates of divisions are independent ofeach other and there is no temporal correlation in their growth. With these assumptions themodel builds a diversified multi-divisional structure. Starting from a single product evolvingK. Matia et al.: Statistical properties of business firms etc. 499100 102 104 106 Firm sales, S10-210-1100 Standard deviation, σ(g|S)β=0.20(a)-10 -5 0 5 10 Scaled growth rate of firms, g/S-0.2010-410-310-210-1100 Probability density, S-0.20P(g|S) small medium large(b)Fig. 1 – (a) Firms are divided into 10 groups according to firm sale S. We find the standard deviationσ(g|S) of the growth rates scales as a power law, σ(g|S) ∼ S−β , with β = 0.20 ± 0.01. Symbolsare data points and the solid line is a regression fit. (b) The PDF of the growth rates for small(S < 102), medium (102 < S < 104), and large (104 < S) values of S is scaled by their standarddeviation. Note the collapse of the histograms of the three groups which confirms the scaling exponentβ. The dashed line is f0 as predicted by the model approximating the results of ref. [5] givenby f0(x) ≈ a0 exp[−a1[1 + 0.75 ln(1 + a2 x2)]][1 + 0.75 ln(1 + a2 x2)]1/2, where a0, a1, and a2 areparameters of the model.to a multi-product firm, this model reproduces a number of empirical observations and makessome predictions which we discuss in detail below along with results and predictions of othermodels which attempt to address the problem of business firm growth.Consider a firm i of sales Si with Ni products whose sales are ξi,j , where j = 1, 2, . . . , Ni.Thus the firm size in terms of the sales of its products is given as Si =∑j=Nij=1 ξi,j . The growthrate isgi(t) ≡ log(Si(t+∆t)Si(t))= logSi(t+∆t)− logSi(t), (1)where Si(t) and Si(t + ∆t) are the sales, in units of 103 British pounds, of firm i beingconsidered in the year t and t + ∆t, respectively. Pharmaceutical data has seasonal effect,and hence the analysis of quarterly data will have effects due to seasonality. To remove anyseasonality, that might be present, we analyze the annual data instead of the quarterly data.Recent studies have demonstrated power law scaling in economic systems [6]. In particular500 EUROPHYSICS LETTERS100 102 104 106 Firm sales, S100102104 Mean sale of products, E(ξ|S)α=0.57(a)10-4 10-2 100 102 104 Scaled product sales, ξ/S0.5710-910-610-3100 Probability density, S0.57 ρ1(ξ|S) small medium large(b)Fig. 2 – (a) Mean, E(ξ|S), of the product sale conditioned for a firm of sale S. We observe that themean scales as E(ξ|S) ∼ Sα with α = 0.57 ± 0.02. Symbols are data points and the solid line isa regression fit. (b) PDF of the product sales for small (S < 102), medium (102 < S < 104), andlarge (104 < S) values of S scaled by S0.57. Note the collapse of the PDFs of the three groups whichconfirms the scaling exponent α.the standard deviation σ of the growth rates of diverse systems including firm sales [6] or grossdomestic product (GDP) of countries [7] scales as a power law of S.The models of refs. [1–3,5,8] all predict that the standard deviation of growth rates amongstall firms with the same sales scales as a power law σ(g|S) ∼ S−β . Further, the model of ref. [5]predicts that the probability density function PDF p(g|S) of growth rates for a size of firm Sscales as a function of S asp(g|S) ∼ 1S−βf0(gS−β), (2)where f0 is a symmetric function of a specific “tent-shaped” form resulting from a convolutionof log-normal distributions and a Gaussian distribution, with parameters dependent on theparameters of the model. Figure 1a plots the scaling of the standard deviation σ(g|S). Weobserve σ(g|S) ∼ S−β with β = 0.19 ± 0.01. Figure 1b plots the scaled PDF as given byeq. (2) for three sales groups; small (S < 102), medium (102 < S < 104) and large (104 < S).The figure also plots f0 as predicted by ref. [5]. The model of ref. [5] further predicts that thePDF of the product size ξ for a fixed firm size S, ρ1(ξ|S) should scale asρ1(ξ|S) ∼ 1Sαf1(ξSα), (3)K. Matia et al.: Statistical properties of business firms etc. 501100 102 104 106 Firm sales, S100101102103 Mean number of products, E(N|S)γ=0.42(a)10-3 10-2 10-1 100 101 102 Scaled number of products, N/S0.4210-410-2100102 Probability density, S0.42 ρ2( N|S) small medium large(b)Fig. 3 – (a) Mean, E(N |S), of the number of products N for a firm of sale S. Symbols are datapoints and the solid line is a regression fit. We observe that the mean scales as E(N |S) ∼ Sγ withγ = 0.42 ± 0.01. (b) PDF of the number of product for small (S < 102), medium (102 < S < 104),and large (104 < S) values of S scaled by S0.42. Note the partial collapse of the PDFs of the threegroups which confirms the scaling exponent γ. For small values of N , which also corresponds to smallvalues of S, the statistics become poor. This statistical errors gets even more amplified when wedivide small values of N by S0.42. Thus we observe poor quality of data collapse for N/S0.42 < 1.The data collapse is better for N/S0.42 > 1, where we have good statistics.where again f1 depends on the parameters of the model. According to the model discussedin refs. [5, 8], f1 is a log-normal PDF. We then evaluate the average product size E(ξ|S) in afirm of size S, defined as E(ξ|S) = ∫ dξρ1(ξ|S)ξ ∼ Sα. Figure 2a plots E(ξ|S); we observeE(ξ|S) ∼ Sα with α = 0.57±0.02. Figure 2b plots the scaled PDF as given by eq. (3) for threesales groups: small (S < 102), medium (102 < S < 104) and large (104 < S). We observe thatfor each of the groups the PDF ρ1(ξ|S) is consistent with a log-normal distribution by notingin a log-log plot the PDF ρ1(ξ|S) is parabolic, which is tested by performing a regression fit.According to ref. [5], the PDF ρ2(N |S) of number of products N in a firm of size S shouldobey the scaling relationρ2(N |S) ∼ 1Sγf2(NSγ), (4)where the function f2 is log-normal and depends on the parameters of the model. We evaluatethe average number of products E(N |S) for a firm of size S. Using eq. (4) we note thatE(N |S) = ∫ dNρ1(N |S)N ∼ Sγ . Figure 3a plots the expectation E(N |S) and we observe502 EUROPHYSICS LETTERS0 2 4 6 8 Sales, log10S10-410-2100 Probability density, P(log10S) all products at launch all firms Wp = 0.88 Wf = 1.72Fig. 4-4 -2 0 2 4 growth10-510-410-310-210-1100 Probability density, P(g) g∆t=1yearg∆t=10 year surrogate dataFig. 5Fig. 4 – PDF of products sales (squares) firm sales (circles) between 1990–2001. The variance ofthe PDFs of products sales at launch and firm sales are estimated to be Wp = 0.88 and Wf = 1.72,respectively. This gives Wf −Wp = 0.84 and β = 0.24 (cf. eq. (7)), which is approximately what isobserved empirically as predicted by [5].Fig. 5 – Probability density function (PDF), P (g∆t), of products for ∆t = 1 year (solid line) and∆t = 10 years (dashed line). Circles represent the surrogate data [9]. In the absence of correlationwe expect the data for ∆t = 10 to coincide with the PDF of the surrogate points.that E(N |S) ∼ Sγ with γ = 0.42± 0.01. Figure 3b plots the scaled PDF ρ2(N |S) as given byeq. (4) for three groups: small (S < 102), medium (102 < S < 104) and large (104 < S).According to [5], the relations between the scaling exponents α, β, and γ are given byγ = 1− α , (5)β =1− α2, (6)which we find to be approximately valid for the PHID database.According to the model discussed in ref. [5], the distribution of product sizes in eachfirm scales with the product size at launch, Sat launch, according to ρ0(ξ/Sat launch), whichis approximately log-normal. Model [5] postulates that the PDF of Sat launch is log-normal,i.e., P (logSat launch) is Gaussian with variance Wp and each firm is characterized by a fixedvalue of Sat launch. Furthermore, ref. [5] predicts that the distribution of firm sales is closeto log-normal, i.e. the PDF P (logS) is Gaussian with variance Wf . With these hypothesesref. [5] derives thatβ =Wf −Wp2 Wf. (7)Figure 4 plots PDF of annual products sales, products sales at launch P (logSat launch), andfirm sales P (logS) between 1990–2001. The variance of the PDFs of products sales at launchand firm sales are estimated to be Wp = 0.88, Wf = 1.72, respectively. This gives β =(Wf − Wp)/2 Wf = 0.24 which is approximately what is observed empirically. We employtwo methods to estimate W (the standard deviation): i) Estimate W from the definition, i.e.W 2 = (1/(N −1))∑i=Ni=1 (xi−〈x〉)2, where {x1, x2, . . . xN} is a set of data and 〈x〉 is the meanof the set {xi}. ii) First estimate the PDF from the set {xi}, then perform a regression fitwith a log-normal function to the PDF. The standard deviation W will be one of the fittingK. Matia et al.: Statistical properties of business firms etc. 503parameter. Hence estimate W from the estimated parameter value from the least-square log-normal fit to the PDF. We observe both this method gives similar values of W and the ratio β(cf. eq. (7)) remains unchanged as long as we consistently use one of the 2 methods describedabove. Our estimate of W presented here is using the former method.Reference [5] postulates products growth rate to be Gaussian and temporally uncorrelated.To test this postulate, fig. 5 plots the PDF P (g) of the growth g of the products ∆t = 1 and∆t = 10 year [9]. We see that the empirical distribution is not growing via random multiplica-tive process as ref. [5] postulates, but has the same tent shape distribution as the distributionof firm sales growth rate, suggesting that the products themselves may not be elementaryunits but may be comprised of smaller interacting subunits. Figure 5 also plots PDF P (g′)surrogate data obtained by summation of the 10 annual growth rates from the empirical distri-bution. We observe that the P (g∆t=10) for products differs from the surrogate data implyingthere are significant anti-correlations in the growth dynamics between successive years.In summary we study the statistical properties of the internal structure of a firm and itsgrowth. We identify three scaling exponents relating the i) sales of the products, ξ, ii) thenumber of products, N , and iii) the standard deviation of the growth rates, σ, of a firmwith its sales S. Our analysis confirms the features predicted in ref. [5]. However, we findthat the postulate of the model, namely, the growth rate of the products is uncorrelated andGaussian, is not accurate. Thus the model of ref. [5] can be regarded as a first step towardsthe explanation of the dynamics of the firm growth.∗ ∗ ∗We thank L. A. N. Amaral and S. Havlin for helpful discussions and suggestions, andNSF and Merck Foundation (EPRIS Program) for financial support.REFERENCES[1] Ijiri Y. and Simon H. A., Skew Distributions and the Sizes of Business Firms (North-Holland,Amsterdam) 1997.[2] Sutton J., Physica A, 312 (2002) 577.[3] Wyart M. and Bouchaud J. P., cond-math/0210479v2 (19 Nov. 2002).[4] Coase R. H., Economica, 4 (1937) 386; Coase R. H., The Nature of Firm: Origins, Evolutionand Development (Oxford University Press, New York) 1993, pp. 34-74; Mansfield E., ResearchPolicy, 20 (1991) 1; Pakes A. and Sokoloff K. L., Proc. Nat. Ac. Sci. USA, 93 (1996) 12655;Erikson R. and Pakes A., Rev. of Eco. Studies, 62 (1995) 53.[5] Amaral L. A. N., Buldyrev S. V., Havlin S., Salinger M. A. and Stanley H. E., Phys.Rev. Lett., 80 (1998) 1385.[6] Stanley M. H. R., Amaral L. A. N., Buldyrev S. V., Havlin S., Leschhorn H., MaassP., Salinger M. A. and Stanley H. E., Nature (London), 379 (1996) 804; Amaral L. A. N.,Buldyrev S. V., Havlin S., Leschhorn H., Maass P., Salinger M. A., Stanley H. E. andStanley M. H. R., J. Phys. I, 7 (1997) 621; Buldyrev S. V., Amaral L. A. N., Havlin S.,Leschhorn H., Maass P., Salinger M. A., Stanley H. E. and Stanley M. H. R., J. Phys.I, 7 (1997) 635; Lee Y., Amaral L. A. N., Canning D., Meyer M. and Stanley H. E., Phy.Rev. Lett., 81 (1998) 3275; Canning D., Amaral L. A. N., Lee Y., Meyer M. and StanleyH. E., Econ. Lett., 60 (1998) 335; Keitt T. and Stanley H. E., Nature (London), 393 (1998)257.[7] Plerou V., Amaral L. A. N., Gopikrishnan P., Meyer M. and Stanley H. E., Nature(London), 400 (1999) 433.[8] De Fabritiis G., Pammolli G. and Riccaboni M., Physica A, 324 (2003) 38.[9] gi(t) ≡ log(Si(t+ 1)/Si(t)). And g∆t=10 =∑10i=1gi, ∆t=1.

Statistical Properties of Business Firms Structure and Growth

We analyze a database comprising quarterly sales of 55624
pharmaceutical products commercialized by 3939 pharmaceutical
firms in the period 1992–2001. We study the probability density
function (PDF) of growth in firms and product sales and find that
the width of the PDF of growth decays with the sales as a power
law with exponent $\beta=0.20\pm 0.01$. We also find that the
average sales of products scales with the firm sales as a power
law with exponent $\alpha=0.57\pm 0.02$. And that the average
number products of a firm scales with the firm sales as a power
law with exponent $\gamma=0.42\pm 0.02$. We compare these
findings with the predictions of models proposed till date on
growth of business firms

K. Matia

S. V. Buldyrev

F. Pammolli

M. Riccaboni

H. E. Stanley

EDP Sciences OAI-PMH repository (1.2.0)

Statistical properties of business firms 
structure and growth

We analyze a database comprising quarterly sales of 55624 pharmaceutical products commercialized by 3939 pharmaceutical firms in the period 1992-2001. We study the probability density function (PDF) of growth in firms and product sales and find that the width of the PDF of growth decays with the sales as a power law with exponent Î² = 0.20 Â± 0.01. We also find that the average sales of products scales with the firm sales as a power law with exponent Î± = 0.57 Â± 0.02. And that the average number products of a firm scales with the firm sales as a power law with exponent Î³ = 0.42 Â± 0.02. We compare these findings with the predictions of models proposed till date on growth of business firms

Statistical Properties of Business Firms Structure and Growth

Abstract

Similar works

Full text

Available Versions

Archivio della ricerca della Scuola IMT Alti Studi Lucca

Crossref

Florence Research

Crossref

EDP Sciences OAI-PMH repository (1.2.0)

IMT Institutional Repository