The kingdom of Syldavia has five economic classes. Feeling one day that there is too great a disparity among the wealth of the classes, the King asks his Finance Minister to devise a plan to redistribute the wealth of the classes. The FM proposes that the wealth of the top two classes be averaged, then the wealth of classes 2 and 3 be averaged, then the wealth of classes 3 and 4 be averaged, and finally that the wealth of classes 4 and 5 be averaged—a four-step process. The King likes the sound of the plan, but suggests that the progression should be from the lowest classes upwards. (In either case, the result will be a four-class system.) Which of the two plans should the lowest class prefer? Why?
This problem may need some clarifying before work can begin. Suppose that the wealth of the five classes is a, b, c, d, and e, with a at the top. Four new classes will be created. If we start at the top, these classes will be M1 through M4:
Similarly, we could express the four levels starting at the bottom.
If we start at the bottom, we’ll have
Now, assuming that the lowest (new) class will be either M4 or M1‘, M4 is clearly better since the wealth at the top is figured in. (This could all be fooled around with on a spreadsheet — see what happens with specific values.)