Adrienne consumes three goods, x, y, and z. Her consumption preference is given bythe utility function:u(x, y, z) = x*y} z#,(Pz-Py, Pz)where x, y, and z are the quantities she consumes (respectively). Let p =be the prices of the goods. Answer the following questions, and justify your answerscarefully and mathematically.(a) Last year p =sumption bundle last year.(1, 10, 10) and Adrienne's income was 35. Find her optimal con-(b) This year, the prices of x and z have both risen 3.5% while the price of y has risen7%, and Adrienne got an income raise of 3%. Find the (approximate) percentagechanges of her consumption in the three goods due to these changes in prices andincome. Does her raise adequately compensate her for the changed prices? Whatadditional percentage change of her income would be needed if we were to restoreAdrienne to last year's exact utility level?

U(x,y,z) = x17y27z47Px = 1Py = 10Pz = 10

Answered: Adrienne consumes three goods, x,y, and…

Exploring Economics

8th Edition

ISBN:9781544336329

Author:Robert L. Sexton

Publisher:Robert L. Sexton

Chapter10: Consumer Choice Theory

Section: Chapter Questions

Problem 8P

See similar textbooks

Similar questions

2. Consider a consumer whose preference relation over the consumption set X = R can be repre- sented by the utility function: u (x1, x2, 13) = A (x1 – T1)ª (x2 – ã2)° (x3 – ã3)" - where A, a, B and y are all strictly positive. (a) Why can you assume A = 1 and a + B+y= 1 without loss of generality? Do so for the rest of the question. (b) Solve the UMP and derive the consumer's Walrasian demand. (c) Find the indirect utility function, the minimum expenditure function, and the Hicksian demand function.
2. A consumer has a utility fuinction given by a) Derive an expression for the two marginal utilities: MU (x1, 22) and MU2 (21, r2). Since AMRS = -YU use these marginal utilities to derive a simple expression for the MRS (r1, 22). b) Optimal choice on the part of the consumer implies MI RS = -. Suppose M 20, p1 = p2 = 1. Show the optimal choice in this case on a well-labelled graph of the budget set. Include an indifference curve consistent with these preferences. c) Now keep income at 20, and pi = 1, but set p2 - 2. Show the optimal choice in this case on a well-labelled graplh of the budget set. Inclnde an indifference curve consistent with these preferences.
3. Suppose Mary enjoys Pepsi and Coke according to the functionU(P;C) = 4C + 5P. What does her utility function say about her MRS of Coke for Pepsi? What do her indi§erence curves look like? What type of goods are Pepsi and Coke for Mary? If Pepsi and Coke each cost $1 and Mary has $20 to spend on these products, how many units of each product should she buy in order to maximize her utility? Show this utility maximiz- ing combination combination of Pepsi and Coke on the graph. how would her consumption and utility maximizing bundle of Coke and Pepsi change if the price of Coke decreases to 50 cents. 4. Vera is an impoverished graduate student who has only $100 a month to spend on food. She has read in a government publication that she can assure an adequate diet by eating only peanut butter and carrots in the Öxed ratio of 2 pounds of peanut butter to 1 pound of carrots, so she decides to limit her diet to that regime. a) If peanut butter costs $4 per pound and carrots cost $2 per…
4. Let u(z) be a utility function that represents the preferences of a household. We say that the function v(r) is a monotonic transformation of u if f(-) is a strictly increasing function and v(z) = f(u(z)). (a) Show that if v(z) is a monotonic transformation of u, it represents the same preforences. (b) Can you explain why taking a monotonic transformation of a utility function does not change the marginal rate of substitution? (c) What kind of preferences are reprosented by a utility function of the form u(a1, 12) = VEi +#2? What about the function v(z1, r2) = 13z + 13r2? (d) Consider the function u(r1, 2) = VE1#2. What kind of preferences does it represent? Is the function (11, r2) jz a monotonic transformation of 1 u(r1, 12)? Is the function w(r1, 12) u(r1, 12)? = zfr a monotonic transformation of
A consumer is making purchases of products Alpha and Beta such that the marginal utility of product Alpha is 63 and the marginal utility of product Beta is 40. The price of product Alpha is $9, and the price of product Beta is $5. The utility-maximizing rule suggests that, to stay within a given budget constraint, this consumer should Multiple Choice O O O increase consumption of product Beta and increase consumption of product Alpha. make no change in the consumption of Alpha or Beta. increase consumption of product Beta and decrease consumption of product Alpha. increase consumption of product Alpha and decrease consumption of product Beta.
No written by hand solution Assume Fred has the following preference relation on [0, 1]: x ≿ y if and only if x ≤ y for all x ̸= 1 and y ̸= 1; and 1 ≻ z for all z ∈ (0, 1), 1 ∼ 1 and 0 ≻ 1. Does there exists a utility representation for this preference relation? If yes, provide a utility function. If no, explain. Show your work.
5. What kind of preferences are represented by a utility function of the form u(x1, x2) = x1 + V*2? Is the utility function v(x1, x2) = xỉ + 2x1Vx2+x2 a monotonic transformation of u(x1, x2)? 6. Consider the utility function u(x1, x2) = Vx1x2. What kind of pref- erences does it represent? Is the function v(1, x2) transformation of u(x1, x2)? Is the function w(x1, x2) = xỉx, a monotonic transformation of u(x1, x2)? = x²x2 a monotonic
Carol needs to decide how to spend her wealth on sh and chicken.For Carol, 1 lb of sh is equivalent to 2 lb of chicken. Her preferencecan be represented by the utility functionu(x; y) = 2x + ywhere x is the quantity of sh (in lbs) and y is the quantity of chicken(in lbs). The consumption set is R2+. (a) Draw two typical indi erence curves for Carol, one correspondingto a utility level of u1 and one corresponding to a utility level u2,where 0 < u1 < u2. Make sure you label the slope of the indif-ference curves and the intercepts with the horizontal and verticalaxes.(b) Suppose the price of sh is $1:5 per lb and and the price of chickenis $1 per lb. Carol has $20 to spend on sh and chicken.(i) Draw Carol's budget set, labeling the slope and the interceptpoints clearly.(ii) How much sh and chicken will Carol choose to purchase? I am looking for an answer to the points (c) and (d) below (c) Suppose now that the price of sh increases to $2 per pound.(i) Draw Carol's budget set,…
Mylie’s total utility from singing the same song over and over is 50 utils after one repetition, 90 utils after two repetitions, 120 utils after three repetitions, 80 utils after four repetitions, 20 utils after five repetitions, and -60 utils after six repetitions, as shown in the table below. a. Write down her marginal utility for each repetition. Instructions: Enter your answers as a whole number. If you are entering any negative numbers be sure to include a negative sign (-) in front of those numbers. Repetitions Total Utility Marginal Utility 0 0 — 1 50 2 90 3 120 4 80 5 20 6 -60
EXERCISE 7 Aisha is considering how to allocate the next 6 hours of her free time. She could choose between leisure (L) and helping her neighbour with the house chores. If she decides to help her neighbour, she is going to get paid at £25 per hour, which she can then spend on her favourite pizza (P). Suppose the price of pizza is £12.50. Aisha's preferences for leisure and pizza are given by the following utility function: U(L, P) = 3L + P. (MU₁ = 3, MUp = 1). a) Write down Aisha's budget equation and draw the corresponding budget line. Clearly label the axes and calculate the coordinates of the points of intersection of the budget line with each axis. b) Calculate Aisha's marginal rate of substitution between leisure and pizza. Explain the concept of MRS and interpret the figure obtained. c) Find Aisha's optimal consumption bundle, both algebraically and graphically. Explain your reasoning. d) Would Aisha's optimal choice change if she could get a discount on her pizza purchases so…
John likes Coca-Cola. After consuming one Coke, John has a total utility of 10 utils. After two Cokes, he has a total utility of 25 utils. After three Cokes, he has a total utility of 50 utils. Does John show diminishing marginal utility for Coke, or does he show increasing marginal utility for Coke? Supposethat John has $3 in his pocket. If Cokes cost $1 each and John is willing to spend one of his dollars on purchasing a first can of Coke, would he spend his second dollar on a Coke, too? What about the third dollar? If John’s marginal utility for Coke keeps on increasing no matter how many Cokes he drinks, would it be fair to say that he is addicted to Coke?
A consumer has the following utility function: Ulx, y) = xy -y, *21 where x and y represents the quantities consumed of goods X and Y. y 20 What will be the substitution and income effects for X and Yassuming that the consumer attempts to maintain the same level of utility achieved before price of Y increased (that is, when price of Y was $1)? SEx= +0.5 IEx = -0.5 SE, = -0.25 IE- = -0.25 SEx= +0.293 IE = -0.293 SEy = -0.414 IE, = +0.414 SEr= +0.25 IE SE, = -0.75 IE, = -0.75 = -0.25 SEx= +0.414 IEx = -0.414 SEy = -0.293 IE, = -0.207 Income = $3 Px= $1, Py= $2

SEE MORE QUESTIONS

Recommended textbooks for you

Exploring Economics

Economics

ISBN:

9781544336329

Author:

Robert L. Sexton

Publisher:

SAGE Publications, Inc

Economics:

Economics

ISBN:

9781285859460

Author:

BOYES, William

Publisher:

Cengage Learning

Microeconomic Theory

Economics

ISBN:

9781337517942

Author:

NICHOLSON

Publisher:

Cengage

Exploring Economics

Economics

ISBN:

9781544336329

Author:

Robert L. Sexton

Publisher:

SAGE Publications, Inc

Economics:

Economics

ISBN:

9781285859460

Author:

BOYES, William

Publisher:

Cengage Learning

Microeconomic Theory

Economics

ISBN:

9781337517942

Author:

NICHOLSON

Publisher:

Cengage

Micro Economics For Today

Economics

ISBN:

9781337613064

Author:

Tucker, Irvin B.

Publisher:

Cengage,

Economics: Private and Public Choice (MindTap Cou…

Economics

ISBN:

9781305506725

Author:

James D. Gwartney, Richard L. Stroup, Russell S. Sobel, David A. Macpherson

Publisher:

Cengage Learning

Microeconomics: Private and Public Choice (MindTa…

Economics

ISBN:

9781305506893

Author:

James D. Gwartney, Richard L. Stroup, Russell S. Sobel, David A. Macpherson

Publisher:

Cengage Learning

SEE MORE TEXTBOOKS