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, … , 𝑥_𝑛 ) = 𝑢(𝑥₁, 𝑥₂, … , 𝑥_𝑛 ) + (𝑢(𝑥₁, 𝑥₂, … , 𝑥_𝑛 ))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

The Correct Option is A, C

The given question requires us to identify which function(s) among the provided options represent the same preference relation as the utility function \( u: R^n_+ \to R_+ \), which is complete, transitive, and continuous over all bundles of \( n \) goods.

To determine this, we have to check whether the functions are monotonic transformations of the utility function \( u \). A particular utility representation is valid if it is a monotonically increasing transformation of another, implying they represent the same preferences.

Let's analyze each option: 

1. The function \( f(x_1, x_2, \ldots, x_n) = u(x_1, x_2, \ldots, x_n) + (u(x_1, x_2, \ldots, x_n))^3 \) is a strictly increasing transformation of \( u \) since the term \((u(x_1, x_2, \ldots, x_n))^3\) is always positive if \( u \) is positive and strictly increases as \( u \) increases. Hence, \( f \) maintains the same preference ordering as \( u \).
2. For the function \( g(x_1, x_2, \ldots, x_n) = u(x_1, x_2, \ldots, x_n) + \sum_{i=1}^{n} x_i \), we see that the addition of \(\sum_{i=1}^{n} x_i\) can potentially change the preference order, especially if \( x_i \) values are significant. It does not depend solely on \( u \) and is not a monotonic transformation, thus altering preferences.
3. The function \( h(x_1, x_2, \ldots, x_n) = (u(x_1, x_2, \ldots, x_n))^{\frac{1}{n}} \) is another form of strictly increasing transformation because if \( u \) is positive and increases, \( (u)^{\frac{1}{n}} \) also increases. Therefore, \( h \) maintains the same preference order represented by \( u \).
4. The function \( m(x_1, x_2, \ldots, x_n) = u(x_1, x_2, \ldots, x_n) + (x_1^2 + x_2^2 + \ldots + x_n^2)^{0.5} \) includes an additional component \((x_1^2 + x_2^2 + \ldots + x_n^2)^{0.5}\), which potentially alters the ordering of preferences unless \( m \) depends solely on \( u \). Here, it represents new preferences based on the sum of squares, violating the original preference relation.

Therefore, the options that preserve the same preference relation as the given utility function \( u \) are:

• \( f(x_1, x_2, \ldots, x_n) = u(x_1, x_2, \ldots, x_n) + (u(x_1, x_2, \ldots, x_n))^3 \)
• \( h(x_1, x_2, \ldots, x_n) = (u(x_1, x_2, \ldots, x_n))^{\frac{1}{n}} \)

 

Both of these functions are derived via monotonic transformations of the original utility function, and hence preserve the same preference relation.