BoSe, Bottazzi-Secchi model of firm growth

Provided the dependencies above are satisfied, the installation should be straightforward. Download the latest available tar.gz package from the cafed repository. Unpack the package

tar xvzf bose-{version}.tar.gz

move inside the source directory

cd bose-{version}

run the configure script, build and install

./configure
make
make install

For more detailed instructions see the INSTALL file in the package.

bose_step1 and bose_step2 are just intermediate utilities written to numerically check the correctness of the analytical analysis on which the model is based.

The program bose and the scripts bose_growth and bose_sizes provide a simulation of the Bottazzi-Secchi models. The latter keeps track of the size evolution of the firms while the formers simply generate the growth rates densities (and several other things…. see the documentation below).

This is the basic program for the generation of Bottazzi-Secchi growth rates distribution. One can provide several command lines options (default values are specified in square brackets):

-N the number of firms                            [100]
-M the average number of micro-shocks per firm    [100]
-D the type of micro-shock distribution           [0]
   0:normal 1:uniform 2:gamma
-d the shape parameter of the Gamma distribution
-m the mean of the micro-shock distribution       [0.0]
-v the variance of the micro shock distribution   [.01]
-T the number of simulation steps                 [1]

The program prints the rate of growth of each firm at each period.

This is essentially the same as bose_growth_ with the only difference that together with the growth rate also the size of each firm is printed at each rime step.

This C program analyzes the growth rates probability distribution \(p(x;M,N)\) defined above.

This program takes as input the parameters N,M and a specification for the density g. Both Gamma and Gaussian distributions are available as micro-shock distribution. However, they are always considered of unit variance and mean zero. The options setting the model parameter are (default values are specified in square brackets):

-n the number of firms             [100] 
-m the number of shocks            [0] 
-t the typeof shocks:              [0]
   0    gaussian
   >0   Gamma distributed, value defines the shape parameter
-N the number of points to plot    [100]

The output of the program is chosen with the option -O

-O set the type of output [0]

  0. growth rates distribution:                          x,F_m(x) 
  1. growth rates density:                               x,f_m(x)
  2. Laplace distribution and density:                   x,F_L(x),f_L(x)
  3. deviation from Laplace distribution:                x,D(x)
  4. max deviation from Laplace and critical sample:     argmaxD(x),maxD(x),N(.5),N(.05),N(.01)

D is the absolute deviation of the density of the model from a Laplace density

\[ D(x) = |F_m(x) - F_L(x)| \]

The parameter \(N()\) printed by the option -O 4 is the effective number of points to discriminate with confidence f the model distribution from the Laplace, using the Kolmogorov-Smirnov statistics. It's the solution of:

\[ (\sqrt{N/2}+.12+.11/\sqrt{N/2}) N(f) = Q_{ks}^{-1} (f) \]

where \(Q_{ks}^{-1}\) is the inverse distribution of the K.-S. statistics.

This Python script is intended to check the convergence toward a Laplace density (i.e. symmetric exponential) of the sum of Gaussian variates whose number follows a geometric distribution.

Notice that for a geometric distribution

\[ \text{Prob}\{x=k\} = (1-a) \, a^k \]

and

\[ \text{Prob}\{x \leq N\} = 1-a^{N+1} \]

so that this variate can be generated by::

\[ x = \left[ \frac{log(1-\epsilon)}{log(a)} \right] \]

where \(\epsilon\) is an uniform distribution in \([0,1)\) and \([x]\) stands for the integer part of \(x\) (i.e. the C function floor()).

This Python script is written to check how the Bose-Einstein partition of M micro-shocks across N firms generates a geometric distribution of shocks for each single firm.