payment system and robustness

[@okaars]

how do get to 1/(1-b)? What expression is on the LHS? dx/dy or dy/dx When i try to differantiate I get a+bx = x+y =>dxb = dx + dy => dy = dx(b-1) => dy/dx = b-1 or dx/dy= - 1/(1-b) What am I doing wrong?

[@HansOlavMelberg]

It is easiest to use the formulat for the sum of a geometric series to derive how much you have to cut from x relative to y: 1/ (1- b) If you differentiate: dy/dx tells you what will happen to the number of y's if you increase x by one unit. dx/dy tells you what will happen to the number of x's if you if you increase y by one unit Both are relevant, but is the latter we are most interested in in our case. Notice that they will both be negative as long as b (the bonus share) is less than one (i.e. less than 100 percent). Since it will always be negative we may just as well report the (absolute) number which is more intuitive. We say "we have to cut 200 dollars." Not "We have to change by negative 200." So, we want the answer to be "how much do we have to cut" which means that the answer has to be in absolute terms. That is why your answer is correct, but you can just ignore the minus sign. This is commonly done in economics when, for instance, we talk about elasticities. Often we say that the price elasticity is 0.2, but we really mean that it is -0.2.

[@okaars]

Hans Olav Melberg Thanks for the explanation Just to clarify; in this setting, dx/dy says how many units of x I don't have to cut if I instead reduce y by 1? In other words, the larger the expression, the more efficient y is at reducing the defecit compared to x.

[@HansOlavMelberg]

Ole Kristian Aars Yes. These are two sides of the same coin. It tells you that if you want to produce more y (eg. quality) then you have to cut x (volume). Which also implies that if you have a deficit, cutting y will be require a smaller cut than decreasing x. By the way, note that the setup is a bit unrealistic (and I do not want to push the story too far), but it may help capture an important mechanism. Also, I think it is easier to think in terms of the geometric series than the derivaties. Here is another video where I explain this approach in more detail: Robust payment systems