Distance to frontier, productivity distribution and travelling waves

Jan Lorenz, Fabrizio Zilibotti, Michael König 19 November 2015



Even within a single country and narrowly defined sectors, numerous empirical studies have documented a large variation in the productivity levels of firms (Bartelsman and Doms 2000). These studies also emphasise that heterogeneity is a robust empirical finding across industries. Hence, at an aggregate level, productivity growth stems from productivity improvements across a broad population of heterogeneous firms. In this sense, growth needs to be understood as a distribution rather than a level effect. Hsieh and Klenow (2009) show that the spread of productivity distribution is a measure of resource misallocation and, as such, is an important determinant of aggregate productivity differences across countries.  

The broad distribution of productivity across firms is related to the different strategies firms follow in order to make productivity improvements. In order to explain this phenomenon, in a forthcoming paper (König et al. 2016) we consider a model economy where differentiated intermediate goods are produced by monopolistically competitive firms. Firms producing different varieties have heterogeneous productivities that change stochastically over time. The key assumption is that there are technological spillovers across firms producing different intermediates. More specifically, a firm producing a certain variety can try to imitate the technology used by a firm producing another variety whenever the opportunity arises. A distinctive feature of their model is that firms make endogenous decisions of whether to undertake in-house research and innovation or to imitate other firms’ technologies. The success of the imitation strategy depends on the availability of better technologies (which in turn depends on the endogenous distribution of productivity) and their absorptive capacities.

The explicit formulation of firms’ research and development behaviour distinguishes our model from previous contributions to the literature. Early contributions focusing on firm size and growth rate distributions like Gibrat (1931), Pareto (1896) and Simon (1955), as well as more recent ones by Fu et al. (2005) and Stanley et al. (1996), do not take into account research and development decisions of firms. Nor does the recent paper by Lucas and Moll (2014) that focuses instead on the trade-off in the use of time between production. In Perla et al. (2014) firms can choose either to produce, or to search for existing technologies to imitate. Other models such as Klette and Kortum (2004), Luttmer (2007) and Acemoglu and Cao (2015) explicitly model firms’ research and development effort decisions but do not incorporate the trade-off incumbent firms face between investing in innovation or copying and adopting technologies from other incumbent firms. Relative to these authors, our theory focuses on the innovation versus imitation trade-off. The prediction of their theory is consistent with the empirical evidence that firms closer to the technology frontier engage in more research and development investments (Griffith et al. 2003), and that large firms spend more on research and development than smaller ones. For example, Mandel (2011) finds that US firms with 5,000 or more employees spend more than twice as much per worker on research and development as those with 100-500 employees.

Productivity distributions as travelling waves

In König et al. (2016), we analyse a large data set containing information about the productivity of western European firms in the period between 1995 and 2003. Their main empirical findings can be summarised as follows:

  • First, the distribution of high-productivity firms is well described by a power law.
  • Second, the distribution of low-productivity firms is also well approximated by a power law, although this approximation is less accurate, arguably due to noisy data at low productivity levels for small firms.

An example of the empirical productivity distribution for French firms can be seen in Figure 1, while the exponents for the estimated power laws are shown in Table 1.

  • Third, the distribution is characterised by a constant growth rate over time, where both the right and the left power law are fairly stable (see Table 1).

This implies that the evolution over time of the empirical productivity distribution can be described as a 'traveling wave'.

While the first property is well known (see Corcos et al. 2007), the second and the third have not been emphasised in the literature.

Figure 1. Total factor productivity

Notes: The left panel shows the total factor productivity (TFP) distribution of French firms over the years from 1995 to 2003 (vertical axes in logarithmic scale). The right-hand panel shows the mean and standard deviation of the log-TFP, with fitted regression lines.

Source: König et al. (2016).

Table 1. Estimated power law exponents

Notes: Estimated power law exponents for the right and left tail of the probability density function, denoted by λ and ρ. The percentage of firms on which the regression is computed is shown as well as the corresponding coefficient of determination R2.

Source: König et al. (2016).

The innovation-imitation channel

Starting from ex ante identical firms, our model generates heterogeneous productivity distributions with power law tails for small and large productivity values, with exponents that match the empirical estimates. A comparison of the theoretical predictions with the empirical observation can be seen in Figure 2. Moreover, as found in the empirical analysis, the theoretical distribution evolves endogenously over time as a 'traveling wave' with stable shape, which means that at any point in time the distribution is the same up to a location parameter.

The endogenous innovation-imitation choice is crucial for this result. If the population of firms consisted of a fixed proportion of innovators and imitators (and no endogenous innovation-imitation choice were allowed), the limiting distribution would feature an ever increasing variance of productivities across firms. The intuition for this result is simple – firms that are close to the frontier carry out fresh innovation, driving the movement of the productivity frontier. Firms lagging behind choose to imitate and the probability of successful imitation is increasing with the distance to the frontier because these firms can draw from a larger pool of existing technologies. This prevents that an ever growing gap emerges between more and less successful firms.

The optimal growth of productivity requires the right mix of innovation and imitation. Both policies that enhance the innovation success probability and those that stimulate the imitation and diffusion of existing technologies can increase the growth rate of the economy. However, while the first leads to an increase in inequality, the latter reduces inequality across firms’ productivities. Interestingly, an economy in which technologies can easily be imitated but in which there is insufficient in-house research and development does not generate growth. Thus, a balanced approach is required, fostering both the capacity of firms to generate innovations in-house and an environment in which these innovations can diffuse throughout the economy.

Figure 2. Comparison of the empirical distributions of Figure 1 

Notes: Comparison of the empirical distributions of Figure 1 with the calibrated model for the years 1995, 1999 and 2003. The empirical productivity values have been binned to produce the histogram shown in the figure, using 11 bins across all observed productivity values.

Source: König et al. (2016).


Our findings have several policy implications:

  • Research and development subsidies and competition policy versus intellectual property rights

Even though entry, exit and reallocation are important determinants of aggregate productivity growth, entry and exit account for only 25% of total productivity growth (Bartelsman and Doms 2000). A successful growth policy should aim not only at the process of innovation (research and development subsidies) and selection across firms (competition policies), but also the determinants of investments in technology adoption among heterogeneous firms (intellectual property rights protection). Growth-enhancing policies should therefore foster both innovation and imitation, ideally with the right mix.

  • Strong versus weak patent protection

While the absence of patent protection can destroy the incentives of firms to invest in research and development, an overly strong patent protection regime can reduce growth by diminishing the positive effects of knowledge spillovers through imitation and diffusion.

  • Large firms are more conducive for innovation

Large firms tend to innovate while small firms tend to favour imitation, as firms closer to the technological frontier have less imitation opportunities and tend to choose in-house research and development more often. Thus, policymakers should let firms grow big so they can act as technological leaders that supply the economy with a continuous stream of innovations.


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Topics:  Productivity and Innovation

Tags:  productivity, innovation, total factor productivity, TFP

Jacobs University Bremen

Tuntex Professor of International and Development Economics, Yale University; CEPR Research Fellow

Senior Research Associate, Department of Economics, University of Zurich