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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 ln2(10)5.30\ln^2(10) \approx 5.30 (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 (QQ) parts, all agents in each qq are approximately the same size sqs_q. This is known as the narrow quantile condition. In this limit, Equation power_sum_size becomes:

SqtSq0=inqsq010D(; nq)nqsq0=1nqinq10D(; nq)M~t\frac{S_{qt}}{S^0_q}=\frac{\sum\limits_i^{n_q} s^0_q 10^{ D(\cdot;\ n_q)}} {n_q s^0_q} = \frac{1}{n_q } {\sum\limits_i^{n_q} 10^{ D(\cdot;\ n_q)} } \equiv \tilde M _t

where D(;nq)D(\cdot; n_q) denotes a draw of nqn_q elements from the distribution D()D(\cdot). The sum represents a sum of all nqn_q elements of this draw. Here, M~t\tilde M_t is introduced as an estimator of the mean of 10D(.)10^{D(.)} computed from the nqn_q values in 10D(.;nq)10^{D(.; n_q)}.

A key idea to consider is that the value of the quantile, represented by M~t\tilde M_t, can be assumed to have been drawn from an underlying distribution called sDs^*_D. This distribution is assumed to have a 'true' mean value MM, for which M~t\tilde M_t is an estimator, and a variance Σ2\Sigma^2. Beyond this, we do not know much about sDs^*_D; it does not need to be normally distributed or have a closed form expression.

The first moment of sDs^*_D (ME[s]M \equiv E[s^*]) serves as a proxy for the levels Sqt/Sq0{S_{qt}}/{S^0_q} shown by quantile qq. To determine MM, we need to consider the limit:

M=limn(1nin10D(; n))=1nn p(t)10tdt= p(t)10tdtE[10D(.)]M= \lim_{n\to\infty} \left( \frac{1}{n} {\sum\limits_i^{n} 10^{ D(\cdot;\ n)} } \right) = \frac{1}{n} \int n \ p(t) 10^t dt = \int \ p(t) 10^t dt \equiv E [10^{D(.)}]

where p(t)p(t) is the probability density function of the distribution D(.)D(.). This is equivalent to: limn(inpi10D(; n))= p(t)10tdt\lim_{n\to\infty} \left( {\sum\limits_i^{n} p_i 10^{ D(\cdot;\ n)} } \right) = \int \ p(t) 10^t dt. This suggests that sqs^{*_q} may be expressed as sq=10D()s^{*_q} = 10^{D(\cdot)} for some D()D(\cdot). If firm-level fluctuations are small enough, the underlying distribution sqs^{*_q} is approximately lognormal.

Keep in mind that the expressions in Equation expectation_10t are a large nn limit. The observed M~t\tilde M_t should converge to these levels progressively as nqn_q 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 D()D(\cdot), 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 σ^\hat \sigma.