# Modelling Investment Opportunities

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

$p(x; k, \theta) = \frac{1}{\Gamma(k)\theta^k} x^{k-1}e^{-\frac{x}{\theta}}$

where $k>0$ is the shape and $\theta>0$ is the scale.  The expected value of a Gamma distributed random variable is $k \theta$ while its variance is $k \theta^2$.

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 $q\in(0,1)$ that will be used to specify the profitability of the economy and defines the cost, $l$, of an opportunity by

$q = \int_0^l p(x; k, \theta)\; dx.$

The attached plot shows the value of $l$ for $k \in (0.2,10.0)$ and $\theta \in (0.2,3.0)$ with $q=0.1$.