I will need to generate random opportunities for the entrepreneurs in my economy. These opportunities will have to be funded, probably by involving other entrepreneurs, and so I need to specify an opportunity by its distributional characteristics and its initial cost.
Typically one would choose a log-normal distribution for the outcome of an investment opportunity, but I don’t like the implied scale invariance of log-normality. I prefer to use the Gamma Distribution, which is sometimes used to model aggregate claims in insurance, with the density given by
where is the shape and is the scale. The expected value of a Gamma distributed random variable is while its variance is .
I need to have a consistent method of defining the initial cost of the opportunity and will base it on the CDF of the distribution, specifically I will define a parameter that will be used to specify the profitability of the economy and defines the cost, , of an opportunity by
The attached plot shows the value of for and with .