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.
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