I'm not sure if you're joking or not by trying to define a quantitative relationship between piracy and stealing, but here goes anyways:
First of all, relatively pricing should not matter. By your logic, if Toyota's price was suddenly 0.5 * price, then stealing just got two times better! What's wrong is wrong.
Second, your equation does not do justice to math. If your definition of X is the same as mine (i.e. average likelihood an individual will buy this item, independent of the world population) then you're saying Toyota's lost income can potentially be GREATER than the price of the car.
In clearer terms, if
dX / d(world population) = 0
then
lost income = X * world population * price
does not have an upper bound! With the explosive nature of the world population, it would seem that theft gets exponentially worse as the years go by.
Where does the logic break down? Well it seems you've modeled the world's purchases of A SINGLE CAR as independent of each other. (In other words, you've modeled the lost income if Toyota had a car stolen by every human in the world.) What you really want, for n people, is
So now the comparison is between (1 - (1-X)^N) and X. In the case of software like Windows 7 where relatively few people will go for a free alternative, X is pretty damn high. I won't argue that it's greater than (1 - (1-X)^N), but it's nowhere near your estimate of 7 billion.
In all seriousness, I definitely agree that the "relative virtue" of digital theft is better than other forms. But, as you acknowledge, there is indeed a loss of value.
First of all, relatively pricing should not matter. By your logic, if Toyota's price was suddenly 0.5 * price, then stealing just got two times better! What's wrong is wrong.
Second, your equation does not do justice to math. If your definition of X is the same as mine (i.e. average likelihood an individual will buy this item, independent of the world population) then you're saying Toyota's lost income can potentially be GREATER than the price of the car.
In clearer terms, if
then does not have an upper bound! With the explosive nature of the world population, it would seem that theft gets exponentially worse as the years go by.Where does the logic break down? Well it seems you've modeled the world's purchases of A SINGLE CAR as independent of each other. (In other words, you've modeled the lost income if Toyota had a car stolen by every human in the world.) What you really want, for n people, is
- and there's our upper bound at P.So now the comparison is between (1 - (1-X)^N) and X. In the case of software like Windows 7 where relatively few people will go for a free alternative, X is pretty damn high. I won't argue that it's greater than (1 - (1-X)^N), but it's nowhere near your estimate of 7 billion.
In all seriousness, I definitely agree that the "relative virtue" of digital theft is better than other forms. But, as you acknowledge, there is indeed a loss of value.