This study examines the firm size distribution of US banks and credit unions. A truncated lognormal distribution describes the size distribution, measured using assets data, of a large population of small, community-based commercial banks. The size distribution of a smaller but increasingly dominant cohort of large banks, which operate a high-volume low-cost retail banking model, exhibits power-law behaviour. There is a progressive increase in skewness over time, and Zipf's Law is rejected as a descriptor of the size distribution in the upper tail. By contrast, the asset size distribution of the population of credit unions conforms closely to the lognormal distribution.
JEL Classification: G21; L10; L16Keywords: Firm Size distribution, Zipf's Law, Gibrat's Law, Banks, Credit Unions __________________________________________________________________________ + The authors gratefully acknowledge the helpful comments of two anonymous referees on an earlier draft of this paper. The usual disclaimer applies.
IntroductionThis study examines the empirical size distribution of US banks and credit unions. It is well known that most empirical firm-size distributions are highly skewed. If firm sizes are subject to proportional random growth, consistent with Gibrat's Law so that log sizes follow random walks, a lognormal cross-sectional firm-size distribution emerges over time (Gibrat, 1931;Sutton, 1997). There is, however, extensive evidence that the lognormal provides a poor approximation to empirical firm-size distributions in the upper tail, which typically exhibit greater skewness than is consistent with lognormality. Certain modifications toGibrat's Law, however, are capable of producing a cross-sectional size distribution that exhibits power-law behaviour. i A strand in the empirical literature examines the application of lognormal and power-law distributions to cross-sectional firm-size data (Simon and Bonnini, 1958;Quandt, 1966;Lucas, 1978;Cabral and Mata, 2003).Pareto (1897) describes the distribution of a collection of N subjects ranked by size, where the density function, denoted f( ), obeys a power law in the upper tail:where X is size, is the size threshold above which the Pareto distribution applies, and is constant. Zipf's Law describes the special case =1 (Zipf, 1949
Estimation MethodLet X i denote the assets of firm i in a particular year, and let x i =ln(X i ). Let X [i] denote the value of the i'th observation when the firms are ranked in descending order of asset size,, and let x [i] =ln(X [i] ). Let k = X [k] denote some threshold value of k that is, initially, assumed to be pre-selected, and let k = x [k] .We examine two candidate distributions for X i : , where is the standard normal density function.For (ii), the likelihood function that would be formed over the two segments of the distribution is discontinuous at the truncation point k . Accordingly, we estimate the parameters k , 2 k , k, k and k in two stages.
Stage 1and sample variance of allwhere is the sta...