Consider a pure exchange economy with two goods, z and y, and two consumers, 1 and 2. The consumers' utility functions are un (r, y) = r²y u₂(x, y) = √x + √y. respectively. The endowments are (3,0) for consumer 1, and (1,1) for consumer 2. You may use without proof the fact that a consumer with utility u(x, y) = xªy has Marshallian demand r(p, w) = ((a+b)p₁² (a+b)p₂) Marshallian demand. and a consumer with utility u₂(x, y) = √√√x + √y has wp₁ P1(P1+P2)¹ P2(P1+P2) (a) Identify the set of all Pareto efficient allocations. Solution: Let (x, y) be the bundle allocated to consumer 1. Feasibility implies that consumer 2 is allocated (4-2,1-y). The marginal rates of substitution are equal if I 2y 4 which holds if and only if r = 4y. Therefore, the Pareto efficient allocations are those that, for some t € [0, 1], give consumer 1 the bundle (4t, t) and consumer 2 the bundle (4-4t, 1-t). (b) Find all Walrasian equilibria. Solution: Let good a be the numeraire and p be the price of good y. Consumer 1 demands (2,1/p) and consumer 2 demands (p, 1/p). The market for good a clears if 2+p=4. Therefore, there is a unique WE given by p= 2, and each consumer getting the bundle (2,1/2). (c) Are your answers to (a) and (b) consistent with the First Welfare Theorem? Solution: Yes, because the allocation in the answer to part (b) is one of the allocations
Consider a pure exchange economy with two goods, z and y, and two consumers, 1 and 2. The consumers' utility functions are un (r, y) = r²y u₂(x, y) = √x + √y. respectively. The endowments are (3,0) for consumer 1, and (1,1) for consumer 2. You may use without proof the fact that a consumer with utility u(x, y) = xªy has Marshallian demand r(p, w) = ((a+b)p₁² (a+b)p₂) Marshallian demand. and a consumer with utility u₂(x, y) = √√√x + √y has wp₁ P1(P1+P2)¹ P2(P1+P2) (a) Identify the set of all Pareto efficient allocations. Solution: Let (x, y) be the bundle allocated to consumer 1. Feasibility implies that consumer 2 is allocated (4-2,1-y). The marginal rates of substitution are equal if I 2y 4 which holds if and only if r = 4y. Therefore, the Pareto efficient allocations are those that, for some t € [0, 1], give consumer 1 the bundle (4t, t) and consumer 2 the bundle (4-4t, 1-t). (b) Find all Walrasian equilibria. Solution: Let good a be the numeraire and p be the price of good y. Consumer 1 demands (2,1/p) and consumer 2 demands (p, 1/p). The market for good a clears if 2+p=4. Therefore, there is a unique WE given by p= 2, and each consumer getting the bundle (2,1/2). (c) Are your answers to (a) and (b) consistent with the First Welfare Theorem? Solution: Yes, because the allocation in the answer to part (b) is one of the allocations
Chapter3: Preferences And Utility
Section: Chapter Questions
Problem 3.9P
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![Consider a pure exchange economy with two goods, z and y, and two consumers, 1 and 2.
The consumers' utility functions are
respectively. The endowments are (3,0) for consumer 1, and (1,1) for consumer 2. You
may use without proof the fact that a consumer with utility u(x, y) = zªy' has Marshallian
demand z(p, w)
Marshallian demand
u₁(x, y) = r²y
u₂(x, y) = √x + √y,
=
((a+b)p₁' (a+b)p₂), and a consumer with utility u2(x, y) = √√x + √√ has
wp₁
P1(P1+P2)¹ P2(P1+P2)
(a) Identify the set of all Pareto efficient allocations.
Solution: Let (r,y) be the bundle allocated to consumer 1. Feasibility implies that
consumer 2 is allocated (4-1,1-y). The marginal rates of substitution are equal if
2y
which holds if and only if x = 4y. Therefore, the Pareto efficient allocations are those
that, for some t € [0, 1], give consumer 1 the bundle (4t, t) and consumer 2 the bundle
(4-4t, 1-t).
(b) Find all Walrasian equilibria.
Solution: Let good r be the numeraire and p be the price of good y. Consumer 1
demands (2,1/p) and consumer 2 demands (p, 1/p). The market for good a clears if
2+p=4. Therefore, there is a unique WE given by p= 2, and each consumer getting
the bundle (2,1/2).
(c) Are your answers to (a) and (b) consistent with the First Welfare Theorem?
Solution: Yes, because the allocation in the answer to part (b) is one of the allocations
in part (a) (corresponding to t-1/2)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa1ab2968-d288-4fd8-b87c-74963c459231%2Fe6fe2e4b-8990-48f9-a07a-50644de823ef%2Flzaxo68_processed.png&w=3840&q=75)
Transcribed Image Text:Consider a pure exchange economy with two goods, z and y, and two consumers, 1 and 2.
The consumers' utility functions are
respectively. The endowments are (3,0) for consumer 1, and (1,1) for consumer 2. You
may use without proof the fact that a consumer with utility u(x, y) = zªy' has Marshallian
demand z(p, w)
Marshallian demand
u₁(x, y) = r²y
u₂(x, y) = √x + √y,
=
((a+b)p₁' (a+b)p₂), and a consumer with utility u2(x, y) = √√x + √√ has
wp₁
P1(P1+P2)¹ P2(P1+P2)
(a) Identify the set of all Pareto efficient allocations.
Solution: Let (r,y) be the bundle allocated to consumer 1. Feasibility implies that
consumer 2 is allocated (4-1,1-y). The marginal rates of substitution are equal if
2y
which holds if and only if x = 4y. Therefore, the Pareto efficient allocations are those
that, for some t € [0, 1], give consumer 1 the bundle (4t, t) and consumer 2 the bundle
(4-4t, 1-t).
(b) Find all Walrasian equilibria.
Solution: Let good r be the numeraire and p be the price of good y. Consumer 1
demands (2,1/p) and consumer 2 demands (p, 1/p). The market for good a clears if
2+p=4. Therefore, there is a unique WE given by p= 2, and each consumer getting
the bundle (2,1/2).
(c) Are your answers to (a) and (b) consistent with the First Welfare Theorem?
Solution: Yes, because the allocation in the answer to part (b) is one of the allocations
in part (a) (corresponding to t-1/2)
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