Question

Suppose that three utility scales u, u¢ and u¢¢ are related as follows: u¢¢ = 3u¢ - 1, and u¢ = 2u + 4. Find the positive linear transformations that convert u¢ into u and u into u¢¢?

Answer 1

Solution:

1. We have uC = 2u + 4, so positive linear transformation converting uC into u can be return as inverse of this function as:

uC = 2u + 4

Subtracting 4 from both sides: uC - 4 = 2u + 4 - 4

Dividing by 2 on both sides: (uC - 4)/2 = 2u/2

So, we get the required as: u = uC/2 - 2

2. To convert u into uCC, we can simply substitute uC (which is in terms of u, into uCC):

uCC = 3*(2u + 4) - 1

uCC = 6u + 12 - 1

uCC = 6u + 11, which is the required linear positive transformation.