«On the Mechanics of Firm Growth∗ Erzo G.J. Luttmer University of Minnesota and Federal Reserve Bank of Minneapolis ABSTRACT The Pareto-like tail of ...»
Federal Reserve Bank of Minneapolis
Research Department Staﬀ Report 440
On the Mechanics of Firm Growth∗
Erzo G.J. Luttmer
University of Minnesota
and Federal Reserve Bank of Minneapolis
The Pareto-like tail of the size distribution of ﬁrms can arise from random growth of productivity or
stochastic accumulation of capital. If the shocks that give rise to ﬁrm growth are perfectly correlated
within a ﬁrm, then the growth rates of small and large ﬁrms are equally volatile, contrary to what is found in the data. If ﬁrm growth is the result of many independent shocks within a ﬁrm, it can take hundreds of years for a few large ﬁrms to emerge. This paper describes an economy with both types of shocks that can account for the thick-tailed ﬁrm size distribution, high entry and exit rates, and the relatively young age of large ﬁrms. The economy is one in which aggregate growth is driven by the creation of new products by both new and incumbent ﬁrms. Some new ﬁrms have better ideas than others and choose to implement those ideas at a more rapid pace. Eventually, such ﬁrms slow down when the quality of their ideas reverts to the mean. As in the data, average growth rates in a cross section of ﬁrms will appear to be independent of ﬁrm size, for all but the smallest ﬁrms.
∗ This paper has evolved from my Federal Reserve Bank of Minneapolis working papers no. 645 (October 2006), no. 649 (January 2007) and no. 657 (February 2008). The data and more detailed proofs are available at www.luttmer.org. I thank Nathalie Pouokam for skillful research assistance. The views expressed herein are those of the author and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System.
1. I Why does the employment size distribution of US ﬁrms look like a Pareto distribution, with the fraction of ﬁrms with more than n employees roughly equal to n−ζ ? Why is the tail index ζ ≈ 1.05 barely high enough for the distribution to have a ﬁnite mean?
More than half of all ﬁrms with any employees have no more than four employees. But there are also almost a thousand ﬁrms with more than ten thousand employees each, and these ﬁrms employ as much as a quarter of the US labor force. What accounts for the large amount of heterogeneity in ﬁrm size? How does this heterogeneity evolve over time? Some benchmark answers to these questions are needed for the systematic use of ﬁrm-level data in the study of aggregate growth and ﬂuctuations.
In the presence of decreasing returns or downward sloping ﬁrm demand curves, it is possible that the highly skewed size distribution entirely reﬂects a highly skewed productivity distribution. Such a productivity distribution can arise if productivity growth is random and only suﬃciently productive ﬁrms can survive. Given iso-elastic cost functions or demand curves, random productivity growth gives rise to Gibrat’s law, which holds that ﬁrm growth rates are independent of size. A stationary size distribution results if employment at incumbent ﬁrms grows more slowly on average than aggregate employment. This distribution has a tail index ζ just above 1 if cost parameters are such that there is only a small gap between entrant and incumbent mean productivity growth rates (Luttmer ).1 This paper associates ﬁrm size not primarily with productivity diﬀerences, but with organization capital (Prescott and Visscher ) that can be accumulated through investment over time. In the model, a ﬁrm produces one or more diﬀerentiated commodities using labor and commodity-speciﬁc blueprints. An entrepreneur can set up a new ﬁrm by producing a start-up blueprint. After that, the ﬁrm can use labor and any of its blueprints to attempt to produce more blueprints for new commodities. Individual blueprints can also become obsolete. The arrival rates of these two types of events are independent and independent across blueprints. Absent other sources of heterogeneity, 1 The ζ = 1 asymptote is known as Zipf’s law. See Axtell  for recent evidence on the ﬁrm size distribution showing that ζ slightly above 1 ﬁts the data well. Well-known empirical studies on Gibrat’s law for ﬁrms, based on growth rate regressions that correct for selection, are Evans  and Hall . Sutton  surveys the literature. Gabaix  uses Gibrat’s law to interpret the city size distribution and contains many useful references on the history of the subject. Rossi-Hansberg and Wright  develop a model of the ﬁrm size distribution in which there are many industries and the ﬁrm size in any given industry follows a stationary process, instead of the non-stationary process implied by Gibrat.
1 this implies that the mean growth rate of a ﬁrm with more than a single blueprint is independent of ﬁrm size–a weak version of Gibrat’s law. Averaging within the ﬁrm implies that the variance of ﬁrm growth is inversely proportional to ﬁrm size, a violation of the strong form of Gibrat’s law according to which the entire distribution of growth rates is independent of ﬁrm size. The economy exhibits balanced growth, and increases in variety add to the aggregate growth rate, as in Romer  and Young . As long as there is entry, the size distribution will be stationary with a right tail that behaves like n−ζ.
Independent within-ﬁrm replication avoids a problem that arises in economies with only ﬁrm-wide productivity shocks. In Luttmer , it takes a standard deviation of ﬁrm employment growth of about 40% per annum to jointly account for the size distribution and the 11% rate of ﬁrm entry observed in the data. This standard deviation is within the range reported by Davis et al.  for all ﬁrms, but implausibly high for large ﬁrms. Here, large ﬁrms are very stable even when small-ﬁrm growth rates are suﬃciently volatile to be consistent with the observed entry and exit rates. In the simplest version of the model, though, this is too much of a good thing and leads to a rather dramatic counterfactual implication: the median age of ﬁrms with more than ten thousand employees is implied to be about 750 years. Stationarity and the weak version of Gibrat’s law force mean incumbent growth rates to be below the growth rate of the aggregate labor force, only about 1% per annum, and averaging within the ﬁrm reduces variance by too much for “lucky” ﬁrms to become large in a relatively short amount of time.
Newly collected data show that the median age of ﬁrms with more than ten thousand employees in 2008 was only about 75 years. With a 40% standard deviation of employment growth, an economy like Luttmer  predicts about 100 years.2 But to account for the relatively young age of large ﬁrms observed in the data, without assuming there is a 30% chance that employment at WalMart will grow or shrink by more than 40% over the next year, requires abandoning Gibrat’s law.
Suppose therefore that some new ﬁrms enter with an initial blueprint of a higher quality than other blueprints in the economy. The resulting higher proﬁts per blueprint create an incentive to copy these blueprints at a higher rate if quality is inherited. If copies stay within the ﬁrm, then these new ﬁrms will grow fast. If a ﬁrm’s quality advantage is transitory, this rapid growth will come to an end eventually. A stationary distribution with a tail index ζ above 1 results if there is positive entry along the balanced 2 A new calibration is available at www.luttmer.org.
2 growth path. A simple formula shows that this tail index will be close to 1 if ﬁrms with high-quality blueprints grow at an equilibrium rate that is slightly below the sum of the growth rate of the aggregate labor force and the hazard rate with which high-quality ﬁrms lose their edge. Thus high-quality ﬁrms can grow fast if the period of rapid growth is not expected to last too long. But there will be variation in how long ﬁrms are in this rapid growth phase, and this variation allows for the appearance of young large ﬁrms. This version of the organization capital interpretation of ﬁrm growth can match the overall size distribution, the amount of entry and exit, as well as the relatively young age of large ﬁrms. Furthermore, although Gibrat’s law does not hold, the mean growth rates of surviving ﬁrms behave like they do in the data: roughly independent of size for most ﬁrms and signiﬁcantly higher for the smallest ﬁrms (Dunne, Roberts and Samuelson ).
Figure I presents some corroborating evidence for the type of histories of ﬁrm growth predicted by the model. It shows the employment histories of 25 of the nearly 1,000 large ﬁrms that had more than ten thousand employees in 2008 (the data are described in Appendix A). The average employment growth rate across all ﬁrms reported in Figure I is almost 18% per annum, and there is considerable variation. In particular, ﬁrm growth rates seem to be much above average when ﬁrms are relatively small, and decline signiﬁcantly when ﬁrms become large. The data shown in Figure I represent only a 3 small sample from a population of slightly under a thousand large ﬁrms. In turn, this population of large ﬁrms was selected over many years from the population of all ﬁrms that were ever set up. In US data, the number of ﬁrms grows at an annual rate about equal to the 1% growth rate of aggregate employment. Combined with an entry rate of 11%, a steady-state calculation implies that the number of ﬁrms that was ever set up is roughly 11 times the current population of around 6 million ﬁrms.3 The thousand or so ﬁrms with ten thousand or more employees are thus a highly selected sample from a universe of about 66 million ﬁrms. In such a selected sample, one might conjecture, it is not surprising to see that large ﬁrms tend to have a history of rapid enough growth to match the age distribution, even though Gibrat’s law holds. The results presented in this paper show that this conjecture is wrong when shocks tend to average out within a ﬁrm. The strings of positive growth needed are too unlikely, and currently active large ﬁrms should be about 750 years old if Gibrat’s law holds.
Related Literature This paper goes back to, interprets, and builds on the type of growth process initially proposed by Yule  and Simon . Yule  was concerned with the number of species in biological genera, and Simon  with word frequencies, city sizes and income distributions. In the context of cities, Krugman [1996, p. 96] described the time it takes for cities to grow large in Simon’s model as an unresolved problem. Simon and Bonini , Ijiri and Simon , and many others since studied ﬁrm growth. Klette and Kortum  describe an economy based on the quality-ladder model of Grossman and Helpman  in which ﬁrm size follows a birth-death process, as in this paper. In their economy, incumbent ﬁrms cannot grow on average because there is a ﬁxed set of commodities and new entrants continuously capture the markets for some of those commodities. This makes it impossible for large ﬁrms to arise. This diﬃculty is resolved here by considering an economy in which the number of commodities can grow over time, as in Romer  and Young . Even without growth in the number of markets, a thick-tailed size distribution can arise in the Klette and Kortum  economy if Gibrat’s law is relaxed along the lines described in this paper.
The models in this paper are highly tractable analytically, and inevitably stylized.
Lentz and Mortensen  use a version of the Klette and Kortum  economy 3 The growth rate of the collection of all historical ﬁrms equals the entry rate times the fraction of all historical ﬁrms that are currently active. In a steady state, it also equals the growth rate of the active number of ﬁrms, which equals the growth rate of aggregate employment.
4 with additional and more ﬂexible sources of heterogeneity. They do not address the thin-right-tail problem but estimate their model using panel data on Danish ﬁrms.4 The Danish ﬁrm size data do not appear to exhibit the striking Pareto shape that is found reliably in U.S. data. The small size of the Danish economy may well account for this– there are as many ﬁrms in the U.S. as there are people in Denmark. When it comes to examining the right tail of the size distribution, a model economy with a continuum of ﬁrms could simply be a better abstraction for the U.S. than for a small country like Denmark. In addition, small countries will have fewer very large ﬁrms if the replication of blueprints across national boundaries or outside language areas comes at additional costs.
Firms in this paper are organizations that operate in (monopolistically) competitive markets and grow through continuous investment in new blueprints, at a level that is proportional to the size of the ﬁrm. One can alternatively view a ﬁrm as a trading post or network in which agents trade repeatedly. Gibrat’s law and the observed size distribution arise if there is population growth and agents search for ﬁrms by randomly sampling other agents and matching with the ﬁrm with which the agent sampled is already matched.
A simple version of such a model is described in Luttmer . Related models of network formation are presented in Jackson  and Jackson and Rogers , and the extensive literature cited therein. Deciding on the relative importance of these alternative interpretations poses diﬃcult identiﬁcation problems.
Outline The economy and its balanced growth path are described in Section 2, together with two alternative formulations of the role of blueprints in production. The stationary size and age distributions are derived in Section 3 and formulas are given for the tail index ζ in the Gibrat and non-Gibrat cases (Propositions 3 and 4), and for the mode of the age distribution of large ﬁrms when both Gibrat’s and Zipf’s law hold (Section 3.5). Calibrations are in Section 4. All proofs and a description of the data are in the appendix.
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