Consider an economy with two consumers A and B. Consumers A and B have utility functions: u(x₁^,x^)=(x^)¹½¹³(x^)¹½² and u(x³, x²)=(x²)¹³(x²)¹². They face prices P₁ and P2 for good 1 and 2 respectively and have income I^ and I³. a) Argue that V(x₁,x^)= ln[u(x^^,x^)] and V(x³,x²)= ln[u(x³, x²)] represent the same preference relationship as u(x₁,x2) and u(x,x) b) Show that consumer A's demand functions for the two goods are: 31A 5p₂ c) Calculate the optimal value of the Lagrange multiplier ^* and explain its meaning d) Determine B's demand functions for the two goods e) We now assume that the consumers have the following initial endowments e^ = (15,1) and e³ = (5,10). Determine the Walrasian equilibrium for this economy, that is i. the equilibrium price (P*, P₂) the equilibrium allocation (x^(p₁, p₂), x^(p₁, p₂)) and (x²³ (p₁, p₂), x² (P₁, P2)) x^(P₁, P₂, I^)= 2⁹ 21A - ;x₁/² (P₁, P₂, I^)=; 5p₁
Consider an economy with two consumers A and B. Consumers A and B have utility functions: u(x₁^,x^)=(x^)¹½¹³(x^)¹½² and u(x³, x²)=(x²)¹³(x²)¹². They face prices P₁ and P2 for good 1 and 2 respectively and have income I^ and I³. a) Argue that V(x₁,x^)= ln[u(x^^,x^)] and V(x³,x²)= ln[u(x³, x²)] represent the same preference relationship as u(x₁,x2) and u(x,x) b) Show that consumer A's demand functions for the two goods are: 31A 5p₂ c) Calculate the optimal value of the Lagrange multiplier ^* and explain its meaning d) Determine B's demand functions for the two goods e) We now assume that the consumers have the following initial endowments e^ = (15,1) and e³ = (5,10). Determine the Walrasian equilibrium for this economy, that is i. the equilibrium price (P*, P₂) the equilibrium allocation (x^(p₁, p₂), x^(p₁, p₂)) and (x²³ (p₁, p₂), x² (P₁, P2)) x^(P₁, P₂, I^)= 2⁹ 21A - ;x₁/² (P₁, P₂, I^)=; 5p₁
Chapter6: Demand Relationships Among Goods
Section: Chapter Questions
Problem 6.9P
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Question
![Consider an economy with two consumers A and B. Consumers A and B have utility
1
functions:
u(x₁^,x^₂)=(x^¹)¹³(x^)¹½² and u(x²,x^²)=(x³)(x²)¹2. They face prices P₁ and P₂
A
3
2
P2
for good 1 and 2 respectively and have income I^ and IB.
A
a) Argue that V(x^,x^)= ln[u(x^,x^)] and V(x³, x²)= ln[u(x³, x²)] represent the same
preference relationship as u(x1,x^)and u(x²,x²)
b) Show that consumer A's demand functions for the two goods are:
A
x^^ (P₁, P₂,I^)= ;x^(P₁, P₂,I^)=
21¹
5p₁
31A
5P₂
*
c) Calculate the optimal value of the Lagrange multiplier and explain its meaning
d) Determine B's demand functions for the two goods
e) We now assume that the consumers have the following initial endowments e^ = (15,1)
and e³=(5,10). Determine the Walrasian equilibrium for this economy, that is
i. the equilibrium price (P,P₂)
A
the equilibrium allocation (x₁¹ (p₁, p₂), x^ (p₁, p₂)) and (x² (p₁, P₂), x² (P₁, P₂))
A
2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7054aeee-8f20-4996-b04f-5b0565ae2a3f%2F3516390b-5819-4b6e-9617-ec2001fb8249%2Fvnhpnis_processed.png&w=3840&q=75)
Transcribed Image Text:Consider an economy with two consumers A and B. Consumers A and B have utility
1
functions:
u(x₁^,x^₂)=(x^¹)¹³(x^)¹½² and u(x²,x^²)=(x³)(x²)¹2. They face prices P₁ and P₂
A
3
2
P2
for good 1 and 2 respectively and have income I^ and IB.
A
a) Argue that V(x^,x^)= ln[u(x^,x^)] and V(x³, x²)= ln[u(x³, x²)] represent the same
preference relationship as u(x1,x^)and u(x²,x²)
b) Show that consumer A's demand functions for the two goods are:
A
x^^ (P₁, P₂,I^)= ;x^(P₁, P₂,I^)=
21¹
5p₁
31A
5P₂
*
c) Calculate the optimal value of the Lagrange multiplier and explain its meaning
d) Determine B's demand functions for the two goods
e) We now assume that the consumers have the following initial endowments e^ = (15,1)
and e³=(5,10). Determine the Walrasian equilibrium for this economy, that is
i. the equilibrium price (P,P₂)
A
the equilibrium allocation (x₁¹ (p₁, p₂), x^ (p₁, p₂)) and (x² (p₁, P₂), x² (P₁, P₂))
A
2
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