Consider a pure exchange economy with two consumers, A and B, and two goods, 1 and 2. Utilities are and and UA(₁, ₂) = 1 +22 UB(₁,12)= 1²1₂. The consumers have endowments A = (0, 2) and B = (5,0). (a) Find all Walrasian equilibria. Solution: The Marshallian demands are (5.0) if Pi < P2. (p, w)=(-t,t) for t € [0,#] if P₁ = P2, (0.=) if Pi > P2. 2w ₂³ (p., w) = (3-3). 3p 3p2, Letting good 2 be the numeraire and writing p= P₁, the endowments are valued at w = 2 and w = 5p. To check market clearing for good 2, consider the three possible cases for A's demand. If p<1, the market clears if 0+5=2, and hence p = 6/5, contradicting the assumption that p<1. Similarly, if p > 1, the market clears if and hence p = 0, contradicting the assumption that p> 1. Finally, if p=1, the market clears if TA Therefore, there is a unique Walrasian equilibrium given by (p = 1,2^¹ = (²+¹), ²² = (0, 3)).
Consider a pure exchange economy with two consumers, A and B, and two goods, 1 and 2. Utilities are and and UA(₁, ₂) = 1 +22 UB(₁,12)= 1²1₂. The consumers have endowments A = (0, 2) and B = (5,0). (a) Find all Walrasian equilibria. Solution: The Marshallian demands are (5.0) if Pi < P2. (p, w)=(-t,t) for t € [0,#] if P₁ = P2, (0.=) if Pi > P2. 2w ₂³ (p., w) = (3-3). 3p 3p2, Letting good 2 be the numeraire and writing p= P₁, the endowments are valued at w = 2 and w = 5p. To check market clearing for good 2, consider the three possible cases for A's demand. If p<1, the market clears if 0+5=2, and hence p = 6/5, contradicting the assumption that p<1. Similarly, if p > 1, the market clears if and hence p = 0, contradicting the assumption that p> 1. Finally, if p=1, the market clears if TA Therefore, there is a unique Walrasian equilibrium given by (p = 1,2^¹ = (²+¹), ²² = (0, 3)).
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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![Consider a pure exchange economy with two consumers, A and B, and two goods, 1 and 2.
Utilities are
UA(#₁, #₂) = ₁ + x₂
³₂.
and UB(₁, ₂) =
The consumers have endowments A = (0, 2) and B=(5,0).
(a) Find all Walrasian equilibria.
and
Solution: The Marshallian demands are
((,0)
if p < P2,
¹ (p, w) = (-t,t) for t € [0,#] if p₁ = P2₁
((0.=)
if pi > P2.
2w
zB (p, w) =
3p
Letting good 2 be the numeraire and writing p = p₁, the endowments are valued at wA = 2
and w³ = 5p. To check market clearing for good 2, consider the three possible cases for
A's demand. If p<1, the market clears if
0+ = 2,
3
302).
3p2,
and hence p = 6/5, contradicting the assumption that p<1. Similarly, if p > 1, the
market clears if
2+
= 2,
and hence p = 0, contradicting the assumption that p> 1. Finally, if p=1, the market
clears if
5
3
Therefore, there is a unique Walrasian equilibrium given by
(p = 1,2^¹ = ( ₁ ) , ²² = ( + )).
2₂+
2.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa1ab2968-d288-4fd8-b87c-74963c459231%2F8709fef2-3c5c-4364-950f-ed32e72a82ef%2Fjp89of_processed.png&w=3840&q=75)
Transcribed Image Text:Consider a pure exchange economy with two consumers, A and B, and two goods, 1 and 2.
Utilities are
UA(#₁, #₂) = ₁ + x₂
³₂.
and UB(₁, ₂) =
The consumers have endowments A = (0, 2) and B=(5,0).
(a) Find all Walrasian equilibria.
and
Solution: The Marshallian demands are
((,0)
if p < P2,
¹ (p, w) = (-t,t) for t € [0,#] if p₁ = P2₁
((0.=)
if pi > P2.
2w
zB (p, w) =
3p
Letting good 2 be the numeraire and writing p = p₁, the endowments are valued at wA = 2
and w³ = 5p. To check market clearing for good 2, consider the three possible cases for
A's demand. If p<1, the market clears if
0+ = 2,
3
302).
3p2,
and hence p = 6/5, contradicting the assumption that p<1. Similarly, if p > 1, the
market clears if
2+
= 2,
and hence p = 0, contradicting the assumption that p> 1. Finally, if p=1, the market
clears if
5
3
Therefore, there is a unique Walrasian equilibrium given by
(p = 1,2^¹ = ( ₁ ) , ²² = ( + )).
2₂+
2.
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