Question

Suppose that the utility function $𝑢: R^𝑛_+→R_+$ represents a complete, transitive and continuous preference relation over all bundles of 𝑛 goods. Then select the choices below in which the function also represents the same preference relation.

• A. 𝑓(𝑥1, 𝑥2, … , 𝑥𝑛 ) = 𝑢(𝑥1, 𝑥2, … , 𝑥𝑛 ) + (𝑢(𝑥1, 𝑥2, … , 𝑥𝑛 ))3
• B. $𝑔(𝑥_1, 𝑥_2, … , 𝑥_𝑛 ) = 𝑢(𝑥_1, 𝑥_2, … , 𝑥_𝑛 ) + ∑^n_{i=1}𝑥_i$
• C. $ℎ(𝑥_1, 𝑥_2, … , 𝑥_𝑛 ) = (𝑢(𝑥_1, 𝑥_2, … , 𝑥_𝑛 )) ^{\frac{1}{n}}$
• D. $𝑚(𝑥_1, 𝑥_2, … , 𝑥_𝑛 ) = 𝑢(𝑥_1, 𝑥_2, … , 𝑥_𝑛 ) + (𝑥^2_1 + 𝑥^2_2 + ⋯ + 𝑥^2_n ) ^{0.5}$

Answer 1

Answer: (A), (C)

𝑓(𝑥1, 𝑥2, … , 𝑥𝑛 ) = 𝑢(𝑥1, 𝑥2, … , 𝑥𝑛 ) + (𝑢(𝑥1, 𝑥2, … , 𝑥𝑛 ))3

Answer 2

The correct options is (A): 𝑓(𝑥1, 𝑥2, … , 𝑥𝑛 ) = 𝑢(𝑥1, 𝑥2, … , 𝑥𝑛 ) + (𝑢(𝑥1, 𝑥2, … , 𝑥𝑛 ))3 and (C): $ℎ(𝑥_1, 𝑥_2, … , 𝑥_𝑛 ) = (𝑢(𝑥_1, 𝑥_2, … , 𝑥_𝑛 )) ^{\frac{1}{n}}$