I am looking for a general way to measure the, for want of a better term, 'efficiency' of some input given some gain (or loss) resulting from the input. I am assuming that there is some brach of economics mathematics that deals with this stuff.
Take a scenario in which different investments can be made in something. One type of investment costs $2 and this investment would increase that something’s value by $10. Another type of investment would cost $5 and this would also increase that something’s value by $10.
In this case, the first investment is a better, more 'efficient' option than the second since less is invested with the same return. This can be represented with 10/2 > 10/5.
Similarly, if one type of investment of $10 resulted in a $100 increase and another type of investment of $10 resulted in a $200 increase, then the latter option is better, and 200/10 > 100/10
So in maximising efficiency we simply look to maximise the numerator and minimise the denominator.
When potential loss is taken into account, the numerator can become negative. But I am not sure how to capture this precisely.
Using the inverse of the above examples, one type of investment costs $2 and results in a loss of $10. The other type costs $5 and would also result in a loss of $10. In this case, the former is better, since both losses are equal but only $2 was spent on the first and $5 on the second. Accordingly, -10/2 < -10/5, so the better option has a numerically lower measure.
With the other example, $10 is invested in both cases and the first investment results in a loss of $100 and the second results in a loss of $200. The first of these is a better option, since the loss is less. But -100/10 > -200/10, so the better option has a numerically higher measure.