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 .

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