On the modeling of size distributions when technologies are complex

Most technologies used nowadays are complex in the sense that the production processes (and products themselves) consist of a large number of components which might interact with each other in complementary ways. Based on this insight, the current paper assumes that the total productivity of any given technology is functionally dependent on the individual productivities of its n components as well as the elasticity of substitution between them, s. Productivities of the components are in turn drawn from certain predefined probability distributions. Based on this set of assumptions, we obtain surprisingly general results regarding the implied cross-sectional distributions of technological productivity. Namely, drawing from the Central Limit Theorem and the Extreme Value Theory, we find that if the number of components of a technology, n, is sufficiently large, these distributions should be well approximated either by: (i) the log-normal distribution – in the case of unitary elasticity of substitution between the components (s=1); (ii) the Weibull distribution – in the case of perfect complementarity between the components (the “weakest link” assumption, s=0), (iii) the Gaussian distribution – in the (empirically very unlikely) case of perfect substitutability between the components (s?8), (iv) a novel “CES/Normal” distribution – in any intermediate CES case, parametrized by the elasticity of substitution between the components (s>0, s?1). We supplement our theoretical results with numerical simulations allowing us to assess the rate of convergence of the true distribution to the theoretical limit with n as well as the dependence of the “CES/Normal” distribution on the degree of complementarity between the technology components, s. Potential empirical applications of the theoretical result – which remain on the research agenda – include providing answers to the following research questions: How well does the “CES/Normal” distribution fit the data on firm sizes, sales, R&D spending, etc.? What is the implied value of s? Do industries seem to differ in terms of their technology complexity as captured by n? Do industries seem to differ in terms of the complementarity of technology components as captured by s? See above See above