Answer all ### 3. Utility RepresentationsFor each of the pairs of utility functions over $X = \mathbb{R}^2_+$, state whether they represent the same preferences. If so, please find a monotone transformation that relates them (i.e., find a function $f : \mathbb{R} \rightarrow \mathbb{R}$ increasing such that $f(U_1(x)) = U_2(x)$ or $f(U_2(x)) = U_1(x)$ for all $x$). If not, find a pair of outcomes which the two utility functions rank differently (i.e., find $x$ and $x'$ such that $U_1(x) > U_1(x')$ and $U_2(x) < U_2(x')$). 1. $U_1(x) = x_1^2 x_2^4$ and $U_2(x) = \frac{1}{2} \ln x_1 + \ln x_2$2. $U_1(x) = x_1 + 2x_2$ and $U_2(x) = x_1^4 + 2x_2^4$3. $U_1(x) = x_1 - 1$ and $U_2(x) = (x_1 - 1)^2$

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ility representations = of the pairs of utility functions over X R2, state whether they erences. If so, please find a monotone transformation that relates ther f: RR increasing such that f(U₁(x)) = U₂(x) or f(U₂(x)) = U₁(x) a pair of outcomes which the two utility functions rank differently (i. at U₁(x) > U₁(x) and U₂(x) < U₂(x')). r) = xx and U₂(x) = ln x₁ + ln x₂ r) = x₁ + 2x₂ and U₂(x) = x² + 2x r) = x₁ - 1 and U₂(x) = (x₁ - 1)²

ility representations = of the pairs of utility functions over X R2, state whether they erences. If so, please find a monotone transformation that relates ther f: RR increasing such that f(U₁(x)) = U₂(x) or f(U₂(x)) = U₁(x) a pair of outcomes which the two utility functions rank differently (i. at U₁(x) > U₁(x) and U₂(x) < U₂(x')). r) = xx and U₂(x) = ln x₁ + ln x₂ r) = x₁ + 2x₂ and U₂(x) = x² + 2x r) = x₁ - 1 and U₂(x) = (x₁ - 1)²

ENGR.ECONOMIC ANALYSIS

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ISBN:9780190931919

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Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Concept explainers

Form Utility

The term utility in general refers to the satisfaction earned from usage of a particular product or service. Utility always plays an important role because it influences various important elements like demand & thereby the price of the good or service as well. Though utility is not easy to measure, various types of economic models can be used indirectly. There can be various types of utilities like economic utility, marginal utility, cardinal utility, form utility.

Question

Answer all

### 3. Utility Representations

For each of the pairs of utility functions over $X = \mathbb{R}^2_+$, state whether they represent the same preferences. If so, please find a monotone transformation that relates them (i.e., find a function $f : \mathbb{R} \rightarrow \mathbb{R}$ increasing such that $f(U_1(x)) = U_2(x)$ or $f(U_2(x)) = U_1(x)$ for all $x$). If not, find a pair of outcomes which the two utility functions rank differently (i.e., find $x$ and $x'$ such that $U_1(x) > U_1(x')$ and $U_2(x) < U_2(x')$).

1. $U_1(x) = x_1^2 x_2^4$ and $U_2(x) = \frac{1}{2} \ln x_1 + \ln x_2$

2. $U_1(x) = x_1 + 2x_2$ and $U_2(x) = x_1^4 + 2x_2^4$

3. $U_1(x) = x_1 - 1$ and $U_2(x) = (x_1 - 1)^2$

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