A consumer has preferences over two goods represented by the utility function u(x₁, x₂) = 21+2√72. (a) Sketch this consumer's indifference curves. Solution: The indifference curves are conver, and are horizontal translations of each other. (b) Find the Marshallian demand function z(p, w).
A consumer has preferences over two goods represented by the utility function u(x₁, x₂) = 21+2√72. (a) Sketch this consumer's indifference curves. Solution: The indifference curves are conver, and are horizontal translations of each other. (b) Find the Marshallian demand function z(p, w).
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
Related questions
Question
please teach explain step by step
![A consumer has preferences over two goods represented by the utility function u(x₁, x2) =
21+2√3₂.
(a) Sketch this consumer's indifference curves.
Solution: The indifference curves are conver, and are horizontal translations of each
other.
(b) Find the Marshallian demand function z(p, w).
Solution: The first-order conditions have a solution with x1, x2 ≥ 0 if w≥. When
we get a corner solution. Putting these together gives
x(p, w) =
if
((-- -· (2) ²) iu≥ #
1 (0, #)
otherwise.
Since preferences are conver, the solution to the first-order conditions is indeed a mazi-
mum.
EXPLAIN:
Suppose that, initially, prices are p = (p₁, p2) and the consumer has wealth w satisfying
w>. If the price of good one doubles while, at the same time, the price of good two
is cut in half, is the consumer made better or worse off (or is it impossible to determine)?
and w>
P2
so the indirect utilities before and after the change are v = = + 2² and v² = ₁ + 2/2 =
P2
Solution: Let p₁ = 2p₁ and p2=2. Since w > SP² we have w >
"
W + respectively. Hence v > v if + 2 > 201 + 1, which is equivalent to
2p1 P2
6p²
w>
P2
by the change in prices.
Find the expenditure function e(p, u) without solving the expenditure minimization
problem.
Since this last condition holds by assumption, the consumer is made worse off](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc5f3af6a-2abe-4d35-9076-d60b28159552%2F0dd5ba98-1097-43d8-924f-5f9a8ff617e5%2Fqxt0egj_processed.png&w=3840&q=75)
Transcribed Image Text:A consumer has preferences over two goods represented by the utility function u(x₁, x2) =
21+2√3₂.
(a) Sketch this consumer's indifference curves.
Solution: The indifference curves are conver, and are horizontal translations of each
other.
(b) Find the Marshallian demand function z(p, w).
Solution: The first-order conditions have a solution with x1, x2 ≥ 0 if w≥. When
we get a corner solution. Putting these together gives
x(p, w) =
if
((-- -· (2) ²) iu≥ #
1 (0, #)
otherwise.
Since preferences are conver, the solution to the first-order conditions is indeed a mazi-
mum.
EXPLAIN:
Suppose that, initially, prices are p = (p₁, p2) and the consumer has wealth w satisfying
w>. If the price of good one doubles while, at the same time, the price of good two
is cut in half, is the consumer made better or worse off (or is it impossible to determine)?
and w>
P2
so the indirect utilities before and after the change are v = = + 2² and v² = ₁ + 2/2 =
P2
Solution: Let p₁ = 2p₁ and p2=2. Since w > SP² we have w >
"
W + respectively. Hence v > v if + 2 > 201 + 1, which is equivalent to
2p1 P2
6p²
w>
P2
by the change in prices.
Find the expenditure function e(p, u) without solving the expenditure minimization
problem.
Since this last condition holds by assumption, the consumer is made worse off
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 6 steps with 1 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Follow-up Questions
Read through expert solutions to related follow-up questions below.
Follow-up Question
how are you able to have cases income >price ratio
Solution
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