The Role of Granularity in the Variance and Tail Probability of Aggregate Output

What drives macroeconomic fluctuations? The recent macroeconomic literature (e.g. Gabaix (2011); Acemoglu et al. (2012)) shows that microeconomic shocks are an important source for macroeconomic fluctuations because of the granularity of an economy. I provide probabilistic results that quantify the role of the granularity in the variance of aggregate output and macroeconomic tail probability. First, in the linear model of Hulten's theorem, I show that the limiting behavior of the variance of aggregate output as the number of firms goes to infinity is equivalent to that of the contribution of the largest firm when the distribution of firm size has a Pareto-like tail. Empirical data in Japan also indicates the significance of the granularity in terms of the variance of aggregate output. Second, I obtain a formula that links the macroeconomic tail probability and the tail probability of microeconomic shocks. Utilizing the formula, I find that the empirical granularity is too low to lead to the large deviation of aggregate output in Japan. Third, I consider non-linear models and obtain the upper bounds for the variance of aggregate output and the macroeconomic tail probability. This shows that without the granularity, macroeconomic fluctuations decay very rapidly as the number of firms increases, which is the diversification argument in the non-linear models. In summary, my results imply that the granularity is a necessary condition both in linear and non-linear models, and that both granularity and non-linear terms are needed to explain the micro-originated large deviation of aggregate output.