Abstract
A robust feature of the corporate growth process is the Laplace, or symmetric exponential, distribution of firm growth rates. In this paper, we sketch out a class of simple theoretical models capable of explaining this empirical regularity. We do not attempt to generalize on where growth opportunities come from, but rather we focus on how firms build upon growth opportunities. We base ourselves on Penrose's (1959) description of firm growth to explain how the interdependent nature of discrete resources may lead to the triggering off of a series of additions to a firm's resources. In a first formal model, we consider the case of employment growth in a hierarchy, and observe that growth rates follow an exponential distribution. In a second model, we include plant and capital as resources and we are able to reproduce a number of stylized facts about firm growth.
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Notes
Reichstein and Jensen [2005] investigate the growth rate distribution of Danish firms, and, unlike most previous work, they observe asymmetries in the growth rate distribution, such that the Laplace is a better fit to the upper tail than the lower tail of the growth rate distribution.
That is, the observed subbotin b parameter (the “shape” parameter) is significantly lower than the Laplace value of one. This highlights the importance of following Bottazzi et al. [2002] and considering the Laplace as a special case in the Subbotin family of distributions.
More in the sense of Arthur [1989] rather than in that of the Kaldor-Verdoorn “dynamic increasing returns.”
For a survey of growth rate autocorrelation, see Coad [2007a].
Here are a few possible examples. Slack may be present because indivisibilities of key inputs may prevent a firm from attaining perfect productive efficiency. Also, slack may creep in as the learning-by-doing effects that increase a worker's productivity are not counterbalanced by increasing demands made of the worker. Furthermore, slack may be necessary because firms must be able to adapt and act flexibly in response to unforeseen contingencies and the changing market environment.
Penrose writes “[a]t all times there exist, within every firm, pools of unused productive services and these, together with the changing knowledge of management, create a productive opportunity which is unique for each firm” [Penrose 1960, p. 2].
See Bak and Chen [1991]; see also Bak et al. [1993] for an economic application. In the “sandpile” model, grains of sand are dropped on top of each other until a sandpile is formed. “[R]andomly dropping on additional sand will result in the slope of the pile increasing to a critical slope, at which point avalanches of all sizes (limited only by the size of the pile) can occur in response to the dropping of a single additional grain of sand” [Bak et al. (1993) p. 7].
Winter writes “routines clearly qualify as resources, given the expansive use of the term ‘resources’ in the literature of the resource-based view … a routine in operation at a particular site can be conceived as a web of coordinating relationships connecting specific resources …” [Winter 1995, pp. 148–149].
We do not need to define the number “large” nor define what happens at the very top of the hierarchy. Also, we do not need to suppose that the number of hierarchies tends to infinity, because we only want to explain the distribution of growth rates for a certain limited range. An implication of this assumption is that this model is not suitable for describing growth processes in very small firms.
Our model is thus in line with the previous theoretical models of industrial structure and dynamics reviewed in Section ‘A discussion of previous models’, where growth opportunities are supposed to arrive by themselves and little attention is paid to their source.
The reader may have noticed that the focus of this example is on growth amounts, rather than growth rates. However, our analysis investigates the potential growth increments for a firm of a given size. In this particular case, therefore, there is a direct correspondence between growth amounts and growth rates.
Our production function emphasizes complementarity in inputs, rather than substitutability. As a result, it bears some similarities to a Leontief production function (which also emphasizes complementarities) while differing from the standard Cobb-Douglas production function (which allows substitutability).
In other words, the result of B DIV rB is the integer obtained when B is divided by rB and any remainder is thrown away. For example, 7 DIV 2 is the integer 3.
If we assume that the conditions of demand are those of perfect competition (i.e. that price is given) then firms face no scale effects and the model in its present form does not rule out infinite increases in size. In this case, it would thus be necessary to impose some restriction that prevents the apparition of cases in which firms can instantly and costlessly experience implausibly large increases in size. To this end one might introduce the existence of some sort of adjustment costs. In the example developed here, however, the demand curve corresponds to the case of imperfect competition. In this paper, which focuses on one lone firm, imperfect competition constitutes the only point of contact of the firm with the rest of the economy.
Cycling though all possible combinations would be extraordinarily computationally intensive. For example, if we allow variables A−E to take any value from 0 to 100, we end up with 1005=1010 combinations. For this reason, we restrict ourselves to “reasonable” values of the inputs. In choosing the initial conditions, we begin by choosing values for the inputs that are relatively close to the values obtained from optimization when the integer constraint is relaxed. We then explore constellations of inputs around this initial value (i.e. localized search) to obtain the initial profit-maximizing input constellation. In subsequent time periods, as demand increases, we apply iterative localized search procedures, bearing in mind that in most cases the corresponding profit-maximizing set of inputs will be “close” (if not equal) to those obtained in the previous step.
See Coad [2009] for a survey.
Note that we do not report the results for growth of profits, because this series has a strong trending component, which comes from the nature of demand growth.
See Coad [2007a] for a survey.
OLS t-statistics that are robust to heteroskedasticity are calculated using the Huber/White/sandwich estimator (i.e. the “robust” option in Stata 10). LAD t -statistics obtained after 1,000 bootstrap replications.
For a survey, see Coad [2009, Chapter 3].
A negative dependence of growth rate variance on size has been found in data of the US manufacturing firms (Amaral et al. [1997]; Bottazzi and Secchi [2003a]), for firms in the worldwide pharmaceutical industry (Bottazzi and Secchi [2006b]), and, to a lesser extent, for French manufacturing Bottazzi et al. [2010]. In the case of Italian manufacturing firms, however, Bottazzi et al. [2007] do not observe any relationship between growth rate variability and size.
While most studies seem to focus on the relationship between (growth of) profit margins and firm growth [Dosi 2007; Coad 2007b; Bottazzi et al. 2008], other studies focus on the interrelationship between growth of profit (amounts) and firm growth [see e.g. Coad forthcoming; Moneta et al. 2010]. Both these streams of research have similar findings.
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Acknowledgements
Thanks go to Giulio Bottazzi, David Brée, Thomas Brenner, Tommaso Ciarli, Mike Dietrich, Giovanni Dosi, Davide Fiaschi, Ricardo Mamede, Luigi Marengo, Rosanna Nisticò, Bernard Paulré, Rekha Rao, Giorgio Ricchiuti, Angelo Secchi, Federico Tamagni, Ulrich Witt, and participants at the “Economic Evolution as a Learning Process” course at the Max Planck Institute, Jena, Germany, March 13–24, 2006; the workshop on “Internal Organisation, Cooperative Relationships among Firms and Competitiveness” in Lucca, January 2007; the Econophysics conference in Ancona, September 27–29, 2007; and the DIME workshop on “production theory” in Pisa, November 8–9, 2010. Three anonymous referees provided many helpful comments, for which I am grateful. Zlata Jakubovic and Katja Mehlis provided excellent research assistance. The usual caveat applies.
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Coad, A., Planck, M. Firms as Bundles of Discrete Resources – Towards an Explanation of the Exponential Distribution of Firm Growth Rates. Eastern Econ J 38, 189–209 (2012). https://doi.org/10.1057/eej.2010.67
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DOI: https://doi.org/10.1057/eej.2010.67