Question

4. Tom has preferences that can be described by the utility function ?=?(?,?)=??+??, ?>0, ?>0, ?≥0, ?≥0, (?+?)≠0

Tom Junior’s preferences can be described by a transformation of Tom’s utility function ?(?), where ?(?)=(?−36)3.

Tom III’s (i.e. Tom’s grandson’s) preferences can be described by a transformation of Tom’s utility function ?(?), where ?(?)=(?−18)4.

4a. Is ?(?) a positive monotonic transformation? Show your work.

What does that imply for Tom and Tom Jr’s preferences? Explain why.

4b. Is ?(?) a positive monotonic transformation? Show your work.

What does that imply for Tom and Tom III’s preferences? Explain why

Answer 1

A positive monotonic transformation is a way to transform a set of numbers into another set of numbers in a such a manner so that their order remains the same. So, G(U(x,y)) will be a positive monotonic transformation of U(x,y) when

G(U(x₁,y₁)) > G(U(x₂,y₂)) if, U(x₁,y₁) > U(x₂,y₂)

To check a positive monotonic transformation , we check   , i.e whether G is a strictly increasing function of U.

4.a) So, we can see   ?(?)=(?−36)3.

  

Tom Junior’s preferences can be described by a positive mono transformation of Tom’s utility function ?. This implies that Tom and Tom Junior have different utility functions but exactly same preferences. So, the indifference curves of their preferences will also be similar but the numbers attached to them will be different.

4.b) We can see ?(?)=(?−18)4

  

Tom III’s (i.e. Tom’s grandson’s) preferences can be described by a transformation of Tom’s utility function U.This implies that Tom III and Tom have different utility functions but exactly same preferences. So, the indifference curves of their preferences will also be similar but the numbers attached to them will be different.