Problem 1
The text said that raising a number to an odd power was a monotonic transformation. What about raising a number to an even power? Is this a monotonic transformation? (Hint: consider the case \(f(u)=u^{2}\).)
Problem 2
Which of the following are monotonic transformations? \((1) u=2 v-13\) (2) \(u=-1 / v^{2} ;\) (3) \(u=1 / v^{2}\) \((4) u=\ln v\) \((5) u=-e^{-v}\) \((6) u=v^{2}\) \((7) u=v^{2}\) for \(v>0 ;(8) u=v^{2}\) for \(v<0\)
Problem 4
What kind of preferences are represented by a utility function of the form \(u\left(x_{1}, x_{2}\right)=\sqrt{x_{1}+x_{2}} ?\) What about the utility function \(v\left(x_{1}, x_{2}\right)=\) \(13 x_{1}+13 x_{2} ?\)
Problem 5
What kind of preferences are represented by a utility function of the form \(u\left(x_{1}, x_{2}\right)=x_{1}+\sqrt{x_{2}} ?\) Is the utility function \(v\left(x_{1}, x_{2}\right)=x_{1}^{2}+2 x_{1} \sqrt{x_{2}}+x_{2}\) a monotonic transformation of \(u\left(x_{1}, x_{2}\right) ?\)
Problem 6
Consider the utility function \(u\left(x_{1}, x_{2}\right)=\sqrt{x_{1} x_{2}}\). What kind of preferences does it represent? Is the function \(v\left(x_{1}, x_{2}\right)=x_{1}^{2} x_{2}\) a monotonic transformation of \(u\left(x_{1}, x_{2}\right) ?\) Is the function \(w\left(x_{1}, x_{2}\right)=x_{1}^{2} x_{2}^{2}\) a monotonic transformation of \(u\left(x_{1}, x_{2}\right) ?\)
