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In this paper

I don't get Preposition 2. In particular firm size can be described as: $$ \DeclareMathOperator{\E}{\mathbb{E}} dS_{it}=0S_{it}+\sigma S_{it}\; dW_{it} $$ or according to the paper $dS_{it}=\sigma S_{it} \varepsilon_{it}$ where $\sigma$ is the same among different firms. So if GDP is given by: $$ Y_t= \sum_{i=1}^N S_{it} $$ And $$ \sigma_{GDP}= \left( \sum_{i=1}^N \sigma^2 \cdot \left(\frac{S_{it}}{Y_t} \right)^2 \right)^{\frac{1}{2}} $$ Where $h=\left[ \sum_{i=1}^N \left(\frac{S_{it}}{Y_t} \right)^2 \right]^{\frac{1}{2}}$ is the square root of the sales herfindahl of the economy. So if we consider an island economy with $N$ firms whose sizes are drawn from a distribution with finite variance and with same volatility (Gibrat's law for variance), as $N \rightarrow \infty$: $$ \sigma_{GDP}= \frac{\E X_t[S_{it}^2]^{\frac{1}{2}}}{\E X_t[S_{it}]} \frac{\sigma}{\sqrt{N}} $$ Second proposition is about the failure of $1/\sqrt{N}$ argument when the firm size is power law distributed. According to the paper if we consider a series of island economies indexed by $N\geq 1$, each economy has $N$ firms whose growth rate volatility is $\sigma$ (Gibrat's law for variance) and whose size $S_1,...,S_N$ are drawn from a power law distribution: $$ P(S>x)=ax^{-\zeta} $$ For $x>a^{\frac{1}{\zeta}}$ with exponent $\zeta \geq 1$, as $N \rightarrow \infty$: $$ \sigma_{GDP}= \frac{\upsilon_{\zeta}}{\ln {N}}\sigma $$ for $\zeta=1$, or: $$ \sigma_{GDP}= \frac{\upsilon_{\zeta}}{N^{1-1/ \zeta}}\sigma $$ for $1< \zeta <2$ or: $$ \sigma_{GDP}= \frac{\upsilon_{\zeta}}{N^{1/2}}\sigma $$ for $\zeta \geq 2$. I don't understand:

  1. How can the process $dS_{it}=\sigma S_{it} \varepsilon_{it}$ be described by a power law distribution;
  2. What is $\upsilon_{\zeta}$, in the paper they say it is a random variable, but I don't get the link with this random variable and the firm size differential equation;
  3. I am trying to implement with Excel a little example of the paper Propositions in order to understand: if i fix $\sigma=12%$ I can simulate $dW_{it}$ with =NORMINV(RAND()) and $dS_{it}$ with =0.12*NORMINV(RAND()). So starting from $S_{i0}=100$ for every $i$ I can simulate $S_i$ at time $t$ for a given number of firms and verify the first preposition. How could I proceed to verify the second proposition, if it is possible, with Excel?
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