Consider an exchange economy with finitely many consumers I and n > 1 goods. Let Z := R¹ and X = Z x R, and suppose each consumer i E I has a utility u: X → R and an endowment w¹ € X. For the purposes of this question, a bundle is any member of X (in particular, consumers are allowed to consume negative quantities of good n). As usual, and allocation is any profile of consumption bundles such that the sum of all consumption is equal to the sum of all the endowments (so what people consume as a group is the same as what they start with as a group). Suppose every agent has preferences that are quasilinear in the nth good, and so has some v¹: R-¹ such that any z¹ = (z¹, x) has u¹(x¹) = v¹(z¹) + xh. Say an allocation = (zª, ™)iet is surplus maximizing if every other allocation ĩ = (zª, ïn)iet has Σier v² (2¹) ≥ Σie1 v² (2¹) (a) Show that, if the allocation is surplus maximizing, then it is Pareto optimal. (b) Show that, if the allocation is not surplus maximizing, then it is not Pareto optimal.
Consider an exchange economy with finitely many consumers I and n > 1 goods. Let Z := R¹ and X = Z x R, and suppose each consumer i E I has a utility u: X → R and an endowment w¹ € X. For the purposes of this question, a bundle is any member of X (in particular, consumers are allowed to consume negative quantities of good n). As usual, and allocation is any profile of consumption bundles such that the sum of all consumption is equal to the sum of all the endowments (so what people consume as a group is the same as what they start with as a group). Suppose every agent has preferences that are quasilinear in the nth good, and so has some v¹: R-¹ such that any z¹ = (z¹, x) has u¹(x¹) = v¹(z¹) + xh. Say an allocation = (zª, ™)iet is surplus maximizing if every other allocation ĩ = (zª, ïn)iet has Σier v² (2¹) ≥ Σie1 v² (2¹) (a) Show that, if the allocation is surplus maximizing, then it is Pareto optimal. (b) Show that, if the allocation is not surplus maximizing, then it is not Pareto optimal.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
Related questions
Question
![1. [This question gives sum kind of foundation for "adding up utilities" as a mea-
sure of efficiency.]
Consider an exchange economy with finitely many consumers I and n > 1
goods. Let Z := R¹ and X = Z × R, and suppose each consumer i E I
has a utility u : X → R and an endowment w E X. For the purposes of this
question, a bundle is any member of X (in particular, consumers are allowed to
consume negative quantities of good n). As usual, and allocation is any profile
of consumption bundles such that the sum of all consumption is equal to the
sum of all the endowments (so what people consume as a group is the same as
what they start with as a group).
Suppose every agent has preferences that are quasilinear in the nth good, and
so has some v¹: R-¹ such that any z' – (z', r'′) has u¹(x¹) = v¹(zª) +. Say
an allocation = (z¹, ™½)ie, is surplus maximizing if every other allocation
I=(2, 2)ier has Eier (2) 2 Σier (2¹)
(a) Show that, if the allocation à is surplus maximizing, then it is Pareto
optimal.
(b) Show that, if the allocation à is not surplus maximizing, then it is not
Pareto optimal.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffd86cda1-af2d-4585-b012-7ab18b7ef9b3%2F6034c7b6-c760-40ee-9612-c5d28abbffda%2Fcwpgm1_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. [This question gives sum kind of foundation for "adding up utilities" as a mea-
sure of efficiency.]
Consider an exchange economy with finitely many consumers I and n > 1
goods. Let Z := R¹ and X = Z × R, and suppose each consumer i E I
has a utility u : X → R and an endowment w E X. For the purposes of this
question, a bundle is any member of X (in particular, consumers are allowed to
consume negative quantities of good n). As usual, and allocation is any profile
of consumption bundles such that the sum of all consumption is equal to the
sum of all the endowments (so what people consume as a group is the same as
what they start with as a group).
Suppose every agent has preferences that are quasilinear in the nth good, and
so has some v¹: R-¹ such that any z' – (z', r'′) has u¹(x¹) = v¹(zª) +. Say
an allocation = (z¹, ™½)ie, is surplus maximizing if every other allocation
I=(2, 2)ier has Eier (2) 2 Σier (2¹)
(a) Show that, if the allocation à is surplus maximizing, then it is Pareto
optimal.
(b) Show that, if the allocation à is not surplus maximizing, then it is not
Pareto optimal.
Expert Solution
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
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
Recommended textbooks for you
Principles of Economics (12th Edition)
Economics
ISBN:
9780134078779
Author:
Karl E. Case, Ray C. Fair, Sharon E. Oster
Publisher:
PEARSON
Engineering Economy (17th Edition)
Economics
ISBN:
9780134870069
Author:
William G. Sullivan, Elin M. Wicks, C. Patrick Koelling
Publisher:
PEARSON
Principles of Economics (12th Edition)
Economics
ISBN:
9780134078779
Author:
Karl E. Case, Ray C. Fair, Sharon E. Oster
Publisher:
PEARSON
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)
Economics
ISBN:
9781305585126
Author:
N. Gregory Mankiw
Publisher:
Cengage Learning
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-…
Economics
ISBN:
9781259290619
Author:
Michael Baye, Jeff Prince
Publisher:
McGraw-Hill Education