The weak preference relations of Bob and Carol can be represented by real valued utility functions
u and v respectively on ? ≡ R2
+. In each of the following cases, indicate whether their weak preference
relations are identical or not. Provide appropriate explanations for your answers.
(i) u(x1, x2) = x1x2 for all (x1, x2) in ?; v(x1, x2)=(x1)3x2 for all (x1, x2) in ?.
(ii) u(x1, x2) = (1 +x1)(1 +x2) for all (x1, x2) in ?; v(x1, x2) = ln(1 +x1) +ln(1 +x2) for all (x1, x2)
in ?.
(iii) u(x1, x2) = x1x2 for all (x1, x2) in ?; v(x1, x2)=1+ x1x2 + (x1)3(x2)3 for all (x1, x2) in ?.