Firm size distributions in an industry with constrained resources

Abstract

We propose an equilibrium model for firm size distribution in an industry with a constrained essential input. The model applies when the population of firms is small and homogeneous and the supply of the necessary input factor is perfectly inelastic. We argue that although the Gibrat assumption obtains, this does not result in the lognormal distribution because of the entries, exits and mergers of firms competing for the inelastic essential resource. Using our own 32-year database of firms, we test the broken-stick, or random-ordered-interval, model that we call the Whitworth distribution, successfully applied by others to a number of data sets, including the abundances of bird species. We propose the Whitworth as the basic model of the equilibrium distribution of firm sizes for such supply-constrained industries, and find it fits our 31-year database best.

Acknowledgements

We thank Tom Callcott, Ruth Callcott, Glen Barnett and an unknown referee for their valuable assistance. We acknowledge the assistance of chairman Ian Farrer and statistician Carol Mische of the Joint Coal Board (now Coal Services Pty Ltd), and Peter Murray in compiling the database.

Notes

¹ See inter alia, Hart and Prais (1956), Simon and Bonini (1958), Ijiri and Simon (1964), Steindl (1965), Quandt (1966), Ijiri and Simon (1974), Ijiri and Simon (1977), Clarke (1979), Audretsch (1995), Sutton (1997), Kwasnicki (1998), Sutton (1998), Axtell (2001) and Gans and Quiggin (2003).

² Tenders were called for a single mining area in 1975, and tenders were not called again until 1990, when one mining area was offered. Single mining areas were offered in 1994, 1996, 1997 and 2001.

³ In general, the expansion of mines by technological improvements is available to all firms, so the proportionate size only changes by the acquisition or disposal of mines (cet. par.)

⁴ Although the resource (NSW black coal) is constrained (is strictly inelastic in supply), the value of NSW coal does not obey Hotelling's (1931) rule (viz. ‘ the price of an exhaustible resource must grow at a rate equal to the rate of interest, both along an efficient extraction path and in a competitive resource industry equilibrium’, Devarajan and Fisher, 1981, p. 66). That is, its value is not determined by its (local) constrained supply, but by the existence of close substitutes: coal from Queensland and foreign suppliers, and natural gas and oil, globally traded.

⁵ Mines are largely homogeneous, being made up of a aggregation of similar units. A single unit of an underground mine can be one ‘continuous miner’ and its associated equipment, and a single mine can be comprised of one or more units.

⁶ This model results from the instantaneous division of the stick, and not from sequential breakages.

⁷ This was first published as a pamphlet in 1898; the first edition of Whitworth's Choice and Chance appeared in 1901 (Manning, 1952). A detailed account of Whitworth's life and work is given in Irwin (1967).

⁸ The term’broken stick’ is sometimes used as a model for an event with an abrupt change (Curnow, 1973; Chui, 1996; Guthrie and Moorhead, 2002), so this can be confusing. Our term ‘Whitworth distribution’ avoids this confusion and pays tribute to its first proponent–‘it [is] lamentable that that … Whitworth … should be receiving no credit for the original derivation of the broken stick formula’ (Ghent and Hanna, 1968). David and Barton (1962, p.23) doubt that Whitworth invented the eponymous distribution: ‘As was customary with the textbook writer of his day, he gives no references’. Whitworth's 1898 pamphlet, however, suggests that this formulation was his (Whitworth, 1901).

⁹ In sequential breaking the magnitude is broken once at a random point; one of the two parts is selected at random and again randomly broken into two parts and so on. In simultaneous breaking the magnitude is broken into the specified number of parts in one action.

¹⁰ Tokeshi (1990), however, showed that the Whitworth (broken-stick) model can result from sequential breakage if the probability of selection for further breakage of a part is positively related to the size of that part.

¹¹ The variance of the distribution of the rth part is given by

where n = the number of parts (divisions) (Barton and David, 1956).

¹² We thank Glen Barnett for coding the program for the tests.

¹³ In 1994 the third largest firm acquired the largest firm, which caused deviation from the Whitworth model. From 1997 on stability returned and the distribution again fitted the Whitworth model. We do not postulate the speed of equilibration, but this observation suggests a period of 1 to 3 years for a sizeable shock to be absorbed. In 2001 there was another major merger, resulting in the ninth largest firm becoming the largest.

¹⁴ From 1970 to 2001, 55 new firms entered the industry.

¹⁵ These distributions are applied to productivity patterns of scientists, word frequency in texts and distribution of articles in journals, respectively. Note that in an earlier paper (Basu, 1992) she used the Whitworth distribution without naming it.

¹⁶ Our analysis is consistent with that of Whittaker (1965), who distinguished between the interpretations of the relative abundance of species in nature in general and sets of interacting species in particular communities, and also with that of King (1964) and Magurran (1988), who claimed that the MacArthur (1957) model applies in narrowly defined communities of taxonomically related organisms.

¹⁷ The existence of many small firms is consistent with Chuang (1999), who found a small optimal firm size for Taiwan manufacturing industries, especially if export oriented.