Journal Article

No. 2010-17 |
May 21, 2010

Investigating the Exponential Age Distribution of Firms

## Abstract

While several plots of the aggregate age distribution suggest that firm age is exponentially distributed, we find some departures from the exponential benchmark. At the lower tail, we find that very young establishments are more numerous than expected, but they face high exit hazards. At the upper tail, the oldest firms are older than the exponential would have predicted. Furthermore, the age distribution of disaggregated industries (such as the international airline industry) is less regular and can display multimodality. Although we focused on departures from the exponential, we found that the exponential was a useful reference point and endorse it as an appropriate benchmark for future work on industrial structure.

## Comments and Questions

This paper won the "Best Junior Paper" Award at the Schumpeter 2010 conference in Aalborg, Denmark, June 21-24.

In Section 4 of the article the author notes (4.1) that "in an economic system characterized by constant entry rates, an exponential age distribution in a cross-section of firms implies a constant survival rate for firms" but later comes to the conclusion (4.3) that the exponential age distribution "does not ...[more]

... hold in the case of very young plants because, although the number of entrants is roughly constant across years, the youngest plants are observed to have a high exit hazard. This stands in contrast to a constant exit hazard over time predicted by the exponential age distribution benchmark."

Survival rates and hazard rates not being constant over time is consistent with the notion that the Weibull distribution is a more accurate representation of survival possibilities than the exponential distribution, a notion which is supported by theory as well as empirically. In fact, it is well known that the exponential distribution is a special case of the Weibull distribution, corresponding to one of its two parameters (the 'shape parameter') being 1. In such a case the two distributions are identical and produce constant hazard rates.

However, fitting Weibull-type equations to survival data typically leads to estimated 'shape' parameters well below 1, indicating higher hazard rates early in the life of firms and lower hazard rates much later. The tail of the Weibull distribution with 'shape' parameter less than unity is simply much fatter than the corresponding exponential distribution. Between the initial high mortality and fat tail late in life implied by a value below 1, the Weibull function may well approximate an exponential function, implying constant hazard rates for intermediate durations.

It is obvious that high initial hazard rates followed by more or less constant rates for a range of durations before a phase of very low hazard rates, as implied by a 'shape' parameter below 1, would result in an age distribution consistent with the author's observations.

Estimating Weibull-type equations on the 1977-2000 data in Table 1 of the article produces 'shape' parameter estimates significantly below 1 for all cohorts except the 1978 and 1984 cohorts, ranging from 0.585 (with standard error 0.008) in 1979 to 0.798 (standard error 0.03) in 1983. Across all 24 cohorts the estimated 'shape' parameter is 0.702 (standard error 0.01), supporting the notion of very high infant mortality in all years except two (1978 and 1984). The results are in line with similar work carried out previously on the basis of another U.S. data set, and also consistent with results for Denmark, Germany, the Netherlands and Portugal.

Empirically and from a theoretical point of view it is better to use the Weibull distribution rather than the exponential distribution to model survival rates. It therefore does not seem appropriate to assume an exponential age distribution for firms as such an assumption would imply constant survival rates (if entry rates are constant). The age distribution of firms should instead be assumed to follow a distribution which implies Weibull-distributed survival rates, at least approximately. It is not inconceivable that assuming a Weibull age distribution would imply a Weibull distribution for survival rates.

Thanks a lot Tomas Brännström, that is a great suggestion that would certainly be worth pursuing.