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«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 ...»

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Federal Reserve Bank of Minneapolis

Research Department Staff Report 440

March 2010

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 [2007]).1 This paper associates firm size not primarily with productivity differences, but with organization capital (Prescott and Visscher [1980]) 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 [2001] 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 [1987] and Hall [1987]. Sutton [1997] surveys the literature. Gabaix [1999] 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 [2007] 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 [1990] and Young [1998]. 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 [2007], 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. [2007] 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 [2007] 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 [1989]).

–  –  –

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 [1925] and Simon [1955]. Yule [1925] was concerned with the number of species in biological genera, and Simon [1955] 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 [1958], Ijiri and Simon [1964], and many others since studied firm growth. Klette and Kortum [2004] describe an economy based on the quality-ladder model of Grossman and Helpman [1991] 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 [1990] and Young [1998]. Even without growth in the number of markets, a thick-tailed size distribution can arise in the Klette and Kortum [2004] 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 [2006] use a version of the Klette and Kortum [2004] 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 [2006]. Related models of network formation are presented in Jackson [2006] and Jackson and Rogers [2007], 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.

2. T E

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