a conjetura on the distribution of firm profit

8/2/2019 A Conjetura on the Distribution of Firm Profit

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of profit

*iKuni Inc., 3400 Hillview Avenue, Building 5, Palo Alto, CA 94304, USA. Ernail:[email protected] am grateful to Julian Wells for explaining his work on firm profits. and an anony-mous reviewer for helpflil criticisms.


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ABSTRACT RESUMEN

mption of political economy isacross firms or sectors tend tooften ~n od el s re formulated in

tendency is assumed to have beent in reality this tendency is never

dist ribution of firm profits is notkcwed to the right. The mode is

less than the mean an d super-profits are present.To understand the distribution of firm profits ageneral probabilistic argument is sketched thatyields a candidate functional form. The overallproperties of the derived distr ibution are clualita-tively consistent with empirical measures, al-though there is more work to be done.

Kc11 ~crorr ls:irms, profit, economic, distribution ,yrobabilistic.

normal de la econonlia politica esde ganancias de las enipresas y deienden a la homogeneidad y a me-los son fornlulados asumiendo qu eexiste. Pero en la realidad esta ten-

a se materializa y la distribution des de las unidad es productivas resul-

ta sesgacla a la derecha . La mediana es menor qu eel yromedio con lo cual aparecen ganancias ex-traordinarias. Para entender la distribucicin d e larentabjlidad d e las empresas un arg umen to ge-neral d c probabiliciad es bosquejado en cste art i-culo con el fin cle obtener una fornla funcionalposible. [,as propiedades gcncrales de la distri-buci6n clerivada son cualitativamente consisten-tes con las estimacioncs enipiricas, au nque a i ~ nclue~ldrabajo yor hacer.

lJr~li~hr.i~sl,~?w:mpresas, ganancias, distribucihnprobabilistica.


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INTRODUCTION

arjoun and Machover [1989], dis-satisfied with the concept of me-chanical equilibrium applied to

" political economy and the con-comitant assumption of a realiseduniform profit rate, outlined aprobabilistic approach to political

economy, which replaced mechanical equilibriumwith statistical equilibrium and a uniform profitrate with a distribution of profit rates. They rea-soned that the proportion of industrialcapital, outof the total capital invested in the economy, whichfinds itself in any given profit bracket will be a pproximated by a gamma distribution, by analogywith the distribution of kinetic energy in a gas atequilibrium. The gamma distribution is a right-skewed distribution. They examined UK indus-try data from 1972 and concluded that it was con-sistent with a gamma distribution. Wells (2001)examined the distributions of profit rates definedin a variety of ways of over 100 000 UK firms andfound right-skewness to be prevalent, but did notinvestigate their functional form. Wright (2004)measured the distribution of firm profits in an

agent-based model of a competitive economy, andfound that the distribution was right-skewed, al-though not well characterised by a gamma distri-bution, even when capital-weighted. Analysis ofthe model suggested that the profit distributionmay be explained by general probabilistic laws.

The remainder of the paper outlines some theo-retical assumptions and derives a candidate func-tional form for the distribution of firm profits.A PROBABILISTIC ARGUMENT

Under normal circumstances a firm expects thata worker adds a value to the product that is boundfrom below by the wage. A firm's markup on costsreflects this value expectation, which may or maynot be validated in the market. Wages are nor-mally paid in installments of between a week andone month, but the markup on costs is validatedin the market at a frequency that depends on therate at which a firm's goods and services are pur-chased by buyers. The frequency of payments toa firm differ widely and depend on the complex-ity of the product and the details of paymentschedules (for example, compare a firm that sellssweets to a firm that sells battleships). The fre-quency mismatch between wage payments andrevenue payments can be mitigated in many dif-ferent ways, not least by the arrangement of capi-tal loans. But whatever the frequency of sale orthe complexity of the product a revenue paymentto a firm partially reflects the value added by thefirm's workers dur ing a period of time. Assumethat the revenue from the sale of a firm's productconsists of a sum of market samples where eachsample represents the value-added by a particu-lar employee working for a small period of time,


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say an hour. Obviously, there are multiple andparticular reasons why a n individual worker add smore or less value to the firm's otal product, mostof which a re difficult to measure, as partially re-flected in the large variety of contested and nego-tiable compensation schemes. Although eachworker normally adds value there is a great dealof local contingency. A worker may be a slackeror a workaholic, an easily replaceable adminis-trator, or a unique, currently fashionable film star.Therefore, the precise value contr ibution of anindividual worker to the product is highly com-plex and largely unknown, particularly when itis considered that the productive cooperation ofmany workers cannot be easily reduced to sepa-rate and orthogonal contributions, as is the casein highly creative industries with production pro-cesses that have yet to mature into separable, re-peatable and well-defined tasks. This local con-tingency and indeterminacy is modelled byassuming that the value-added per worker-houris a random unriable. Consider that a worker i addsa monetary value, Xi, to a firm's product for ev-ery hour worked, where each Xi is an indepen-dent and identically distributed (iid) random vari-able, with mean pX nd variance 0 % The addedvalue is assumed to be globally idd to reflect thecommon determinants of the value-creating powerof a n hour of work, but also random to model lo-cal contingencies. Negat ive Xi represents nega-tive value-added, corresponding to cases in whichthe worker's labour reduces the value of inputs,for example the production of unwanted goods,or a slower than average work pace, an d so forth.

Assume that the distribution of Xi s such thatthe Central Limit Theorem ( C L T )may be applied.Consider a single firm that sets in motion a totalof n worker-hours dur ing a single year. The firm's

total value-added, S n , may therefore be approxi-mated by a normal distribution S, = z;=, Xi =N(np,, no :). The CLT approximation will improvewith the size of the firm, but even for small firmsthe number of iid draws is large given the statedassumptions.

In reality the productivity of workers withinfirms is correlated. For example, employees offirms that employ state-of-the-art machinery, orare exceptionally well-organised, will all tend toad d more value than employees of firms tha temploy out-of-date machinery or ar e badlyorganised. Although competitive processes tendto homogenise the value-added per worker, newinnovations never cease, so that at any momentin time the employees of particular firm will bemore or less productive than the average.A moreaccurate representation of value-added is ob-tained if each Xi s considered to be drawn from adistribution indexed by the firm that employsworker i, at the expense of a considerable increasein model complexity. However, the correlation ofvalue-added within a large firm, which employsdiverse skills and machinery to produce a varietyof products, will be weak. Although a huge mul-tinational is normally considered a single entityfor the purpose of reporting profits, in reality itsets into a motion a large sample of different kindsof labours utilising different kinds of machineryand tools. Hence, for large firms the assumptionthat Xi is sampled from a single, economy-widedistribution is a reasonable approximation, forsmall firms less so. An advantage of modellingvalue-added per worker as a random variable isthat it is possible that total value-added by a firm,S,, is much higher or lower than the norm, butthis event has low probability. The assumption ofa single distribution that determines the value-


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added per worker is able to approximate the di-verse productivities of individual firms.Each worker costs a certain amount to employ

during the year. This cost includes the wage, thecost of inputs used by the worker, the cost of wearand tear on any fixed capital, the cost of rent, lo-cal taxes and so forth, all of which may be differ-ently reported due to local accountancy practices.Again, there is a great deal of contingency. Hencecosts per worker-hour are also modelled as a ran-dom variable. Assume that a worker i costs amonetary value, Yi, o productively employ perhour worked, where each Yi is an idd randomvariable with mean pyand variance a:. This costincludes both the wage and capital costs perworker, and therefore effaces the distinction be-tween variable and constant capital. Costs perworker-hour are also correlated at the firm level:the employees of different firms productivelycombine a greater r r lesser amount of capital. Amore accurate representation of costs would there-fore consider the distribution of constant capitalacross firms conditional on local circumstances,such as firm size, but this extension is not pur-sued here. The assumption that cost per worker-hour is statistically unifrom across firms is anapproximation, which, as for the case of value-added, improves with firm size, under the as-sumption of a tendency toward homogenisationdue to competitive pressures.

Assume that the distribution of Yi is such thatthe CLT may be applied. Hence a firm that sets inmotion n worker-hours during a year has totalcosts that may be approximated by a normal dis-tribution, K I1=C:=, Yj= N(np , ,n o t ) .This approxi-mation also improves with the size of the firm.

Different firms employee different numbers ofworkers and hence the amount of hours worked

for each firm during a year will vary. Define theprofit, P,l, of a firm that sets in motion n hours oflabour in a single year as the ratio of value-addedto costs, P,, = S J K , , , and assume that S, an d Knare independent. P, is the ratio of two normalvariates. Its probability density function (pdt) mayderived by the transformation method (or alter-natively see (Marsaglia, 1965) to give:

where

Equation (1) is the pdf of the rate-of-profit of afirm conditional on n, the number of hours workedfor the firm per year.

Axtell (2001) analysed US Census Bureau datafor US firms trading between 1988 and 1997 andfound that the firm size distribution, where sizeis measured by the number of employees, fol-lowed a special case of a power-law known asZipf's law, and this relationship persisted fromyear to year despite the continual birth and de-mise of firms and other major economic changes.


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During this period the number of reported firmsincreased from 4.9 million to 5.5 million. Gaffeoet al. (2003) found that the size distribution offirms in the G7 group over the period 1987-2000also followed a power-law, but only in limitedcases was the power-law actually Zipf. Fuijiwaraet al. (2004) found that the Zipf law characterisedthe size distribution of about 260 000 large firmsfrom 45European countries during the years 1992-2001. A Zipf law implies that a majority of smallfirms coexist with a decreasing number of dispro-portionately large firms. Firm sizes theoreticallyrange from1 a degenerate case of a self-employedworker) to the whole available workforce, repre-senting a highly unlikely monopolisation of thewhole economy by a single firm.

Theempirical evidence implies that at any pointin time the firm size distribution follows a power-law, and that this distribution is constant, despitethe continual churning of firms in the economy(birth, death, shrinkage and growth). Firms hireand fire employees, and therefore the number ofhours worked for a firm during a year dependson its particular historical growth pattern. To sirn-plify, assume that the average number of employ-ees per firm per year also follows a power-law.This approximation is reasonable if the growthtrajectories of firms do not fluctuate too widelyduring the accounting period. Assume also thatevery employee works the same number of hoursin a year, which is a reasonable simplification.Thefirm hours per year is therefore a constant mul-tiple of the number of firm employees. Firms withmore employees proportionately set in motionmore hours of labour. A constant multiple of a power-law variate is also a power-law variate. Hence,the firm size distribution has the same power-lawform whether firm size is measured by employ-

ees or by the total number of hours worked byemployees.The unconditional rate-of-profit distribution

can therefore be obtained by considering that thenumber of hours worked for a firm during a yearis a random variable N distributed according to aPareto (power-law) distribution:

where a is the shape and P the location param-eter. Assume that firm sizes range between m,hours, which represents a degenerate case of aself-employed worker who trades during the year,to m, hours, which represents a highly unlikelymonopolisation of all social labour by a singlehuge firm (m, >>m,). The truncated Pareto distri-bution

wheref 7 1 ) = F ' ( n )

is formed to ensure that all the probability massis between m, an d m,. Assume that m, is large sothat the discrete firm size distribution can be ap-proximated by the continuous distribution gN.

By the Theorem of Total Probability the un-conditional profit distribution fp(p) is given by:

Expression (2) defines the gN(n)parameter-mixoffp@ I N = n). The rate-of-profit variate is there-fore composed of a parameter-mix of a ratio ofindependent normal variates each conditional ona firm size n, measured in hours per year, distrib-


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uted according to a power-law. Writing (2)in fullyields the pdf of firm profit:

This distribution has 7 parameters: (i) F,, themean value-added per worker-hour, (ii) a;, thevariance of value-added per worker-hour, (iii)p,,the mean cost per worker-hour, (iv)02,, he vari-ance of cost per worker-hour, (v) a, the Paretoexponent of the firm size power-law distribution,where size is measured in worker-hours per year,(vi) m,, he number of hours worked by a singleworker in a year, and (vii)m , he total number ofhours worked in the whole economy during ayear. Both percentage profit, R = 100P, and thegrowth rate of capital invested, G = I + P, aresimple linear transforms of this distribution.

The parameters can be estimated from eco-nomic data and the resulting distribution com-pared to empirical rate-of-profit measures, undervarious simplifying assumptions about how profitis defined (e.g. see Wells, 2001).A good fit wouldimply that the assumptions made in the theoreti-cal derivation are empirically sound. Alterna-tively, best-fit parameters may be directly esti-mated from empirical data, for example by themethod of maximum likelihood estimation, todetermine how well the theoretical distribution

can fit a set of empirical distributions. A good fitcompared to other candidate functional formswould imply that a parameter-mix of a ratio ofnormal variates with parameters conditional ona power-law captures some essential structure ofthe determinants of firm profit, but it would notvalidate the theoretical derivation.

Equation (3) s difficult to analyse so numeri-cal solutions are employed. Figure 1graphs somerepresentative numerical samples of the distribu-tion. The samples range from sharply peaked sym-metrical curves, in which most of the probabilitymass is concentrated about the mode, to lesspeaked distributions that are skewed to the right.Wells' (2001) variety of profit measures yielddistributions that share these characteristics, andtherefore there is qualitative agreement betweenthe theory and the empirical data. But clearly afull quantitative analysis is required.

Figure 2 graphs a sample of fp(p) in log-logscale. The approximate st raight line in the tail isthe signature of a power-law decay of the prob-ability of super-profits. Super-profit outliers arefound in the empirical data, although it has notbeen investigated whether they decay as an ap-proximate power-law.

Further analysis of the pdf fp(x). is required.But the qualitative form of the distribution is suf-ficiently encouraging to consider it a candidatefor fitting to empirical profit measures and forcomparison with other candidate functionalforms. To go beyond models that assume arealised uniform profit rate it is necessary to in-vestigate empirical data on firm profit and pro-pose theoretical explanations of its distribution.This paper is a tentative step in that direction.


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Figure 1Representative numerical samples of the probability density function fp(p)

(a ) Samples plotted on separate scales

(b) The same samples plotted on a singlescale.


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Figure 2A sample of fP(p) plotted in log-log scale

Note the long power-law tail

CONCLUSION

A general probabilistic argument suggests that theempirical rate-of-profit distribution will be con-sistent with a parameter-mix of a ratio of normalvariates with means and variances that dependon a firm size parameter that is distributed ac-cording to a power law.

REFERENCES

Axtell, Robert L. (2001). "Zipf distribution of U.S.firm sizes", Science, 293:1818-1820.

Fa rjoun, Emmanuel a nd Moshe Machover (1989).Laws of Chaos, a Probabilistic Approach to Politi-cal Economy, Verso, London.

Fujiwara, Yoshi, Corrado Di Guilmi, HideakiAoyama, Mauro Gallegati and Waturu Souma.

(2004). "Do Pareto-Zipf and Gibrat laws holdtrue? An analysis with European firms". PllysicaA, 335:197-216.

Gaffeo, Edoardo, Mauro Gallegati and AntonioPalestrini (2003). "On the size distribution offirms: additional evidence from the G7 coun-tries", Physica A, 324:117-123.

Marsaglia, George (1965). "Ratios of normalvari -ables and ratios of sums of uniform variables",Jour nal of tlze Arrzericnn Stati stic al As soci ation ,60(309):193-204.

Wells, Julian (2001). "What is the distribution ofthe rate of profit?", in IWGVT mini-conferenceat the Eastern Economic Association, NewYork, NY.

Wright, Ian (2004). "The social architecture of capi-talism", submitted for publication, preprint athtt p:/ / xxx.lanl.gov/ PS-cache/ cond-mat/pdf/ 0401/ 0401053.pdf

a conjetura on the distribution of firm profit

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