2 Utility representations Recall that for a finite outome set X, the rank-score function U# (r) = #({x' = X : x ≥ x'}) is a utility function representation for any preference relation. In this problem we'll consider the case of a countably infinite outcome set X = {x¹, x², x³, ...}. 1. Write down a preference relation over X for which the rank-score function is a utility representation. Write down another preference relation for which it is not a utility representation. 2. Consider the preference relation over X defined by xix iff i ≤j. Write down a utility representation which assigns non-positive utility to every outcome. Write down another utility representation which assigns non-negative utility to every outcome.

2 Utility representations Recall that for a finite outome set X, the rank-score function U# (r) = #({x' = X : x ≥ x'}) is a utility function representation for any preference relation. In this problem we'll consider the case of a countably infinite outcome set X = {x¹, x², x³, ...}. 1. Write down a preference relation over X for which the rank-score function is a utility representation. Write down another preference relation for which it is not a utility representation. 2. Consider the preference relation over X defined by xix iff i ≤j. Write down a utility representation which assigns non-positive utility to every outcome. Write down another utility representation which assigns non-negative utility to every outcome.

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

See similar textbooks

Related questions

Q: (a) Consider an agent whose preferences over any couple (1, 2), where x₁ € R+ and 2 € R+, e.g.,…

A: The utility is the total satisfaction derived by the consumption of goods. U=f(X,Y) It denotes that…

Q: Suppose that consumer has the following utility function: U(X,Y) = XªY® where 1 > a > 0 and 1 > b>0…

A: In economics, the marginal rate of substitution (MRS) is the quantity of one good that a customer is…

Q: Total utility (utils) 0 25 45 60 70

Q: marginal utility for custard (U.). What is Amelia's marginal rate of substitution (MRS)? Give a…

A: Different utility levels of consumers are taken to be independent. This means that utility obtained…

Q: a. U(x,y) = 10 √ √xy. b. U(x, y) = xy. c. d. Which of the following utility functions represent the…

A: “Since you have posted multiple questions, we will provide the solution only to the first question…

Q: Suppose a consumer has utility function U(X, Y) = X2/3Y1/3. Suppose the consumer has M = $90 to…

A: Utility function : U = X2/3 Y1/3Income : M = 90 Price of X : Px = 15 Price of Y :Py = 6

Q: Which of these utility functions represent the same preferences as (x, y) = x0.5 y0.5 ? a. U(x, y) =…

A: utility function is the relationship between utility and goods consumed. The utility function can be…

Q: Explain how an optimal decision of a consumer can be achieved by using utility-maximizing theory and…

A: In a market, consumer satisfaction can be computed using two types of economic approaches that are…

Q: Consider a consumer who like combination a = (10,1) better than combination b= (3, 3) and is…

A: A utility function is a thinking used in economics to signify how humans or buyers assign values or…

Q: A consumer's preferences are represented by the following utility function: u(x, y) = 2/T+ Vg+7 1.…

A: Utility derived from (x3, y3) bundle will be: u(7, 6)=27+6+7u(7, 6)=14.74 Similarly, utility derived…

Q: 4. The CES utility function is given by U (x, y) = * (i). Show that the first-order conditions for a…

A: Hey champ, Welcome to this platform. Here you will get the answer with better quality in minimum…

Q: Consider a consumer with utility function u(x1, x2) = min(4 min{x1, x2}, x1 + x2} (a) Draw…

A: This can be described as the concept that provides the graphical representation of the combination…

Q: 4. Derive an equation for the indifference curve through the bundle (x2, y2) = (4,9). With an…

A: The graphical showcasing of consumers choices of two given goods which will provide the consumer…

Q: 4) State which well-behaved preference assumptions are violated by: a. indifference curves with…

A: The economics as a study is based upon the rationale of scarcity. Under this, the economies, and the…

Q: 2. A consumer has a utility fuinction given by a) Derive an expression for the two marginal…

A: utility function is an important concept that measures preferences over a set of goods and services.…

Q: Which of the following statements captures the meaning of transitivity of preferences? a. If A is…

A: Transitivity of preferences implies that the consumer is able to rank all available bundles of goods…

Q: Mr. Dogan has 4000 TL to spend. He considers buying meat whose price is 400 TL per kg. and cheese…

A: Budget Constraint: The cap on spending is defined by income. Usually expressed as an inequality to…

Q: ate a table and graphs (a) Suppose we have preferer indifference curve through the bundle X = 30 and…

A: Utility function : U = 10 x2/3 y1/3 Budget Constraint : Px*x + Py*y = M Optimal consumption is…

Q: Can this preference relationship represented by a utility function? Show all details of your claim…

A: Let's evaluate whether the given preference relationship can be represented by a utility…

Q: 4.11 CES utility The CES utility function we have used in this chapter is given by U(x, y) =+. a.…

A: b) For showing the Cobb-Douglas case: When δ=0xy=pxpy1/(0-1)=pypxx=pypxyNow putting the value of x…

Q: 1. For each of the following scaling functions for a von Neumann-Morgenstern utility func- tion,…

A: The von Neumann-Morgenstern utility function highlights the utility while also including the risks…

Q: Mr. MacGregor's preferences are such that he is indifferent between the following bundles of (x,y):…

A: Answer: Given, Bundles: (x,y): (1,4), (2,2), (4,1) U = ln(xy) Let us find the utility at each…

Q: 4. Consider the general multiplicatively separable utility function given by u (x1,x2) = pu1 (x1) u2…

A: Given Utility function U=μX1X2Budget consraint=P1X1+P2X2=WW= income of the consumerP1= price of good…

Q: A consumer has preferences over 3 goods represented by the utility function u(x₁, x2, x3) =…

A: u(x1,x2,x3) = (x1x2 )0.5- 2-x3 Good 1 and 2 are perfectly divisible and Good 3 are invisible and…

Q: For the case of two goods, give an example of a utility function U(x1, x2) that represents the…

A: In financial matters, the utility function estimates the government assistance or fulfillment of a…

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

2 Utility representations
Recall that for a finite outome set X, the rank-score function U# (x) = #({x' = X : x ≥ x'})
is a utility function representation for any preference relation. In this problem we'll
consider the case of a countably infinite outcome set X = {x¹, x², x³, ...}.
1. Write down a preference relation over X for which the rank-score function is a utility
representation. Write down another preference relation for which it is not a utility
representation.
2. Consider the preference relation over X defined by x¹ ≥ x¹ iff i ≤ j. Write down a
utility representation which assigns non-positive utility to every outcome. Write down
another utility representation which assigns non-negative utility to every outcome.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1: Define utility

VIEW

Step 2: Find the preference relations

VIEW

Solution

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 3 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, economics and related others by exploring similar questions and additional content below.

Similar questions

Q12 please help quick Consider a utility function for apples and bananas. You are given the following three bundles: (12, 4), (4, 12), and (8, 8). Under the standard assumptions of the well-behaved preference relationship. you should strictly prefer (8, 8) to (12, 4) and (4, 12). Briefly explain why?
. Answer all parts (a) (c) of this question. (a) Consider an agent whose preferences over any couple (x1, x2), where 2₁ ER+ and x2 € R+, e.g., apples and oranges, is such that she prefers the bundle that is closest to having the same number of apples and oranges. Write a utility function u: R² → R+ which represent these preferences. A politician remarks "Our recent increases in the wage rates of teachers has been a total suc- cess! The shortage of teachers has been reduced drastically. Another, similar wage increase should eliminate this shortage entirely" (b) Explain and illustrate in a diagram what is meant by "income effects" and "substitution effects" of a wage rate change. (c) Explain and illustrate how you would model the labour supply decision of a potential teacher. Do you agree that the wage increase will increase the labour supply in this case? Carefully outline the assumptions underlying your argument.
Liwei consumes X₁ and x2 together in fixed proportions. She always consumes 2 units of X₁ for 1 unit x₂. One utility function that describes her preferences is: u(x1, x2) = 2x1 + x2. u(x1, x2) = min{2x1, x₂}. O u(x₁, x₂) = min{x₁/2, x₂}. u(x1, x2) = 2x1 + 4x2. O u(x1, x2) = 2x1x2.
Exercise 4. In the lectures we have been working the the CRRA utility function, u : R+ which we defined, for any n≥0, as → R, Prove that u(x): -{ 21--1 1-n if ŋE [0,1) U (1,∞) In a if n = 1. 2--1 lim == Inz. →1 1-7 (7) Hint: Use L'Hôpital's Rule. Exercise 5. Consider the following two-period optimization problem we studied in class: max u (x1)+ẞu (x2), s.t. x1+px2=y, (1.22)ER² (8) where u is given by (7), x; represents consumption of good i = {1,2}, ẞ € [0, 1] is the discount factor, p = R++ is the relative price of good 2 in terms of good 1, and y = R++ is income expressed in terms of good 1. For the following two questions, consider an arbitrary n > 0. 1. Find the bundle (1, 2) that solves (8). 2. Notice that the solution (1, 2) depends on the parameters (n,3,p,y), which the con- sumer takes as given. Derive the expressions for ǝr/Op and ǝx/Əp.
Problem 2.3 Let the consumer's preferences over consumption bundles (x1, x2) be represented by the utility function u(x1,X2) = /x1 +1·(x2+1). Which of the following statements is correct? (a) The utility function v,(x1,x,) = In(x, + 1) + 3 ln(x, + 1) also represents the consumer's preferences. (b) The utility function v,(x1,x2) = In(x1 + 1) + In(x2 + 1)² also represents the consumer's preferences. (c) None of the statements is correct.
Consider a consumer with utility function u(x1, x2) = min{4 min{x1, x2}, x1 + x2} (a) Draw indifference curves passing through points (2, 2), (1, 2) and (4, 2). Make sure you correctly determine kink points. What properties of the preferences can you deduce from the shape of indifference curves? (b) When X = R2 +, does UMP have a solution when p1 = p2 = 0? What property of the preference relation did you use to get your answer? (c) Assume that prices are such that p1, p2 2 0 and that p` > 0 for some `e {1, 2} (i.e. the price of at least one good is positive). Derive Walrasian demand. What are the prices for which Walrasian demand is single-valued?
3.