Question

Consider the utility function u¹(x₁, x₂, x₃) = x²_1,x³₂ x₃ .

Find a monotone transformation of u¹(x₁, x₂, x₃) such that the new utility function is of the form u³(x₁, x₂, x₃) = a In x_{1 +} b In x₂ + c In x_3.

Answer 1

Such monotonic transformation can be done so that the powers are added up to 1

We see that

$2+3+1=6\\and\\a=\frac{2}{6}=\frac{1}{3}\\b=\frac{3}{6}=\frac{1}{2}\\c=\frac{1}{6}$

After monotonic transformation of $U^i(X_1,X_2,X_3)$ the new utility function is $U^2(X_1,X_2,X_3)=X_1^{\frac{1}{3}}, X_2^{\frac{1}{2}} ,X_3^{\frac{1}{6}}$