Imagine I have a company that makes widgets, where each widget costs me A dollars to make. Each month I can allocate money toward research and development with the aim of finding a new process that will allow me to build widgets for a cost of A/B dollars. Presume that I know that for each C dollars I spend on research and development there's a D% chance of finding a breakthrough. Of course, spending money on research and development means that I have less to spend on building widgets.
I have a monthly budget of E dollars. This budget is not directly tied to my profit margin, but it is safe to say that it my profit margins influence future budgets (i.e., if I make no widgets for three straight months b/c I do all research and development, it's likely that my budget will be reduced, whereas if I discover a breakthrough the first month my profits will skyrocket and I'll likely see that budget grow over time).
In case that is too abstract, here's the real world scenario I'm interested in solving (although I'd like a more general approach, as well):
- A = 15 dollars
- B = 3
- C = 5 dollars
- D = 2.75%
- E = 30 dollars
That is, today widgets cost me 15 dollars to build but if I can find a breakthrough I know I can make them at 1/3 the cost (5 dollars). For each 5 dollars I spend on research and development there is a 2.75% chance I'll find the breakthrough. However, I have only 30 dollars to spend each month. If I spend it all on research and development and have no success then I have made no widgets for sale. If I spend it all on widget construction I have no chance of finding a breakthrough.
Is there some statistical distribution or formula that can let me plug in these variables and see some sort of breakdown that gives me an idea of whether it's a good idea to spend any money on research and development each month and, if so, how much?