Suppose that there are three goods, good X, good Y, and good Z, and suppose that the consumer has the utility function denoted by U(X, Y, Z) where X, Y, Z denote the quantities of goods. The prices of the goods are denoted by px, px, and pz, respectively, and the income of the consumer is denoted by I. a. Write down the consumer's optimization problem (formally, as we wrote in the lectures). b. Apply the Lagrangian method. Find the necessary conditions (NC) for the consumer's optimization problem of allocating budget between consumption of three goods (good X, good Y, good Z). c. Provide at least two interpretations of the NC (explain what these conditions mean in words). d. What if there were n goods listed as {X₁, X2, ..., Xn}? What would be the necessary condition for the optimal quantities of any pair of goods X₁ and X₁?
Suppose that there are three goods, good X, good Y, and good Z, and suppose that the consumer has the utility function denoted by U(X, Y, Z) where X, Y, Z denote the quantities of goods. The prices of the goods are denoted by px, px, and pz, respectively, and the income of the consumer is denoted by I. a. Write down the consumer's optimization problem (formally, as we wrote in the lectures). b. Apply the Lagrangian method. Find the necessary conditions (NC) for the consumer's optimization problem of allocating budget between consumption of three goods (good X, good Y, good Z). c. Provide at least two interpretations of the NC (explain what these conditions mean in words). d. What if there were n goods listed as {X₁, X2, ..., Xn}? What would be the necessary condition for the optimal quantities of any pair of goods X₁ and X₁?
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
Related questions
Question
![Suppose that there are three goods, good X, good Y, and good Z, and suppose that
the consumer has the utility function denoted by U(X, Y, Z) where X, Y, Z denote
the quantities of goods. The prices of the goods are denoted by px, px, and pz,
respectively, and the income of the consumer is denoted by I.
a. Write down the consumer's optimization problem (formally, as we wrote in the
lectures).
b. Apply the Lagrangian method. Find the necessary conditions (NC) for the
consumer's optimization problem of allocating budget between consumption of
three goods (good X, good Y, good Z).
c.
Provide at least two interpretations of the NC (explain what these conditions
mean in words).
d. What if there were n goods listed as {X₁, X2,..., Xn}? What would be the
necessary condition for the optimal quantities of any pair of goods X; and X;?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5d3b433f-c139-402b-a0b4-3489fb573214%2F653ba3d7-30fa-40aa-959c-d51e0908f8d3%2Fruqwn92h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Suppose that there are three goods, good X, good Y, and good Z, and suppose that
the consumer has the utility function denoted by U(X, Y, Z) where X, Y, Z denote
the quantities of goods. The prices of the goods are denoted by px, px, and pz,
respectively, and the income of the consumer is denoted by I.
a. Write down the consumer's optimization problem (formally, as we wrote in the
lectures).
b. Apply the Lagrangian method. Find the necessary conditions (NC) for the
consumer's optimization problem of allocating budget between consumption of
three goods (good X, good Y, good Z).
c.
Provide at least two interpretations of the NC (explain what these conditions
mean in words).
d. What if there were n goods listed as {X₁, X2,..., Xn}? What would be the
necessary condition for the optimal quantities of any pair of goods X; and X;?
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
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.Recommended textbooks for you
![ENGR.ECONOMIC ANALYSIS](https://compass-isbn-assets.s3.amazonaws.com/isbn_cover_images/9780190931919/9780190931919_smallCoverImage.gif)
![Principles of Economics (12th Edition)](https://www.bartleby.com/isbn_cover_images/9780134078779/9780134078779_smallCoverImage.gif)
Principles of Economics (12th Edition)
Economics
ISBN:
9780134078779
Author:
Karl E. Case, Ray C. Fair, Sharon E. Oster
Publisher:
PEARSON
![Engineering Economy (17th Edition)](https://www.bartleby.com/isbn_cover_images/9780134870069/9780134870069_smallCoverImage.gif)
Engineering Economy (17th Edition)
Economics
ISBN:
9780134870069
Author:
William G. Sullivan, Elin M. Wicks, C. Patrick Koelling
Publisher:
PEARSON
![ENGR.ECONOMIC ANALYSIS](https://compass-isbn-assets.s3.amazonaws.com/isbn_cover_images/9780190931919/9780190931919_smallCoverImage.gif)
![Principles of Economics (12th Edition)](https://www.bartleby.com/isbn_cover_images/9780134078779/9780134078779_smallCoverImage.gif)
Principles of Economics (12th Edition)
Economics
ISBN:
9780134078779
Author:
Karl E. Case, Ray C. Fair, Sharon E. Oster
Publisher:
PEARSON
![Engineering Economy (17th Edition)](https://www.bartleby.com/isbn_cover_images/9780134870069/9780134870069_smallCoverImage.gif)
Engineering Economy (17th Edition)
Economics
ISBN:
9780134870069
Author:
William G. Sullivan, Elin M. Wicks, C. Patrick Koelling
Publisher:
PEARSON
![Principles of Economics (MindTap Course List)](https://www.bartleby.com/isbn_cover_images/9781305585126/9781305585126_smallCoverImage.gif)
Principles of Economics (MindTap Course List)
Economics
ISBN:
9781305585126
Author:
N. Gregory Mankiw
Publisher:
Cengage Learning
![Managerial Economics: A Problem Solving Approach](https://www.bartleby.com/isbn_cover_images/9781337106665/9781337106665_smallCoverImage.gif)
Managerial Economics: A Problem Solving Approach
Economics
ISBN:
9781337106665
Author:
Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike Shor
Publisher:
Cengage Learning
![Managerial Economics & Business Strategy (Mcgraw-…](https://www.bartleby.com/isbn_cover_images/9781259290619/9781259290619_smallCoverImage.gif)
Managerial Economics & Business Strategy (Mcgraw-…
Economics
ISBN:
9781259290619
Author:
Michael Baye, Jeff Prince
Publisher:
McGraw-Hill Education