Consider an economy with two consumers A and B. Consumers A and B have utility1functions:u(x₁^,x^₂)=(x^¹)¹³(x^)¹½² and u(x²,x^²)=(x³)(x²)¹2. They face prices P₁ and P₂A32P2for good 1 and 2 respectively and have income I^ and IB.Aa) Argue that V(x^,x^)= ln[u(x^,x^)] and V(x³, x²)= ln[u(x³, x²)] represent the samepreference relationship as u(x1,x^)and u(x²,x²)b) Show that consumer A's demand functions for the two goods are:Ax^^ (P₁, P₂,I^)= ;x^(P₁, P₂,I^)=21¹5p₁31A5P₂*c) Calculate the optimal value of the Lagrange multiplier and explain its meaningd) Determine B's demand functions for the two goodse) 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 isi. the equilibrium price (P,P₂)Athe equilibrium allocation (x₁¹ (p₁, p₂), x^ (p₁, p₂)) and (x² (p₁, P₂), x² (P₁, P₂))A2

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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

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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