Expected Level of Parts' Time Series
The strategy is to first determine the expected value of the ratio in Equation power_sum_size. This will simplify the characterization of its volatility. The variance of this ratio is approximately proportional to the variance of the log levels by a factor (see Section 5.5).
Expected Level of Parts' Time Series
To proceed, note that from Equation power_sum_size, if the total is split into enough () parts, all agents in each are approximately the same size . This is known as the narrow quantile condition. In this limit, Equation power_sum_size becomes:
where denotes a draw of elements from the distribution . The sum represents a sum of all elements of this draw. Here, is introduced as an estimator of the mean of computed from the values in .
A key idea to consider is that the value of the quantile, represented by , can be assumed to have been drawn from an underlying distribution called . This distribution is assumed to have a 'true' mean value , for which is an estimator, and a variance . Beyond this, we do not know much about ; it does not need to be normally distributed or have a closed form expression.
The first moment of () serves as a proxy for the levels shown by quantile . To determine , we need to consider the limit:
where is the probability density function of the distribution . This is equivalent to: . This suggests that may be expressed as for some . If firm-level fluctuations are small enough, the underlying distribution is approximately lognormal.
Keep in mind that the expressions in Equation expectation_10t are a large limit. The observed should converge to these levels progressively as increases. Mandelbrot comments, "The population moments of a lognormal or approximate lognormal will eventually be approached, but how rapidly? The answer is: 'slowly'." This convergence can be evaluated graphically in Figure Elog_mu0_fnq_0, (equivalent to Figure E9-2, op. cit.). Before discussing the convergence patterns in Figure Elog_mu0_fnq_0, let us introduce some formal tools.
Although there may not be closed form expressions for the fluctuations , these tent shapes (see Figure fl_dist) can naturally be expressed as mixtures of possibly asymmetric log-normal, log-Laplace, or even fatter tail distributions. The log-normal and log-Laplace distributions will be used as clear benchmarks for other more general log-distributions that may appear empirically and for which we do not have an expression. In all experiments, the empirical distribution of micro shocks shows results with features between the log-Laplace and log-normal functions, thus justifying this choice. The conventions for defining the log-normal and log-Laplace distributions are in the Appendix. They are defined such that the theoretical standard deviation of log shocks matches the parameter .