2. General Equilibrium.consumers, each with the same Cobb-Douglas preferences except with differ-ent parameters. Consumer 1 has utility function u(x, x)= (x})"(x})!-«, whileConsumer 2 has utility function u(x, x;) = (xP(x)-P. The endowment ofgood j owned by consumer i is denoted w. The price of good 1 is p, and theprice of good 2 is 1. In the superscript, we denoted the consumer i = 1,2; in thesubscript, we denote the good j= 1,2.Consider an exchange economy with twoWrite the maximisation problem faced by each consumer i =(a)1,2, taking care to define the objective function and the budget constraint.Set up the Lagrangian and find the first order conditions.(b)For each consumer i = 1,2 , use the first-order conditions todetermine the demand functions for each consumer i = 1,2 and for eachgood j = 1,2, in terms of the price p.(c)Find the aggregate demand for each good j = 1,2 and clear themarkets for each good. Hence, show that the equilibrium price pi is givenby the expressionaw; +Bw;[(1-a)w} +(1-B)w}]Define Walras' here and show that this holds here.

The two consumers are i = 1 and 2For consumer 1, the utility function is : For consumer 2, the…

2. General Equilibrium. Consider an exchange economy with two consumers, each with the same Cobb-Douglas preferences except with differ- ent parameters. Consumer 1 has utility function u(x,x) = (x¹)(x₂)¹", while Consumer 2 has utility function u(x², x2) = (x3)(x2). The endowment of good j owned by consumer i is denoted w. The price of good 1 is p, and the price of good 2 is 1. In the superscript, we denoted the consumer i = 1,2; in the subscript, we denote the good j = 1,2. (a) Write the maximisation problem faced by each consumer i = 1,2, taking care to define the objective function and the budget constraint. Set up the Lagrangian and find the first order conditions. (b). (c) For each consumer i = 1,2, use the first-order conditions to determine the demand functions for each consumer i = 1,2 and for each good j = 1,2, in terms of the price p₁. Find the aggregate demand for each good j = 1,2 and clear the markets for each good. Hence, show that the equilibrium price p, is given by the expression aw+Bw²₂ P₁ [(1-a)w+(1-B)w²] Define Walras' here and show that this holds here.

2. General Equilibrium. Consider an exchange economy with two consumers, each with the same Cobb-Douglas preferences except with differ- ent parameters. Consumer 1 has utility function u(x,x) = (x¹)(x₂)¹", while Consumer 2 has utility function u(x², x2) = (x3)(x2). The endowment of good j owned by consumer i is denoted w. The price of good 1 is p, and the price of good 2 is 1. In the superscript, we denoted the consumer i = 1,2; in the subscript, we denote the good j = 1,2. (a) Write the maximisation problem faced by each consumer i = 1,2, taking care to define the objective function and the budget constraint. Set up the Lagrangian and find the first order conditions. (b). (c) For each consumer i = 1,2, use the first-order conditions to determine the demand functions for each consumer i = 1,2 and for each good j = 1,2, in terms of the price p₁. Find the aggregate demand for each good j = 1,2 and clear the markets for each good. Hence, show that the equilibrium price p, is given by the expression aw+Bw²₂ P₁ [(1-a)w+(1-B)w²] Define Walras' here and show that this holds here.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter3: Preferences And Utility

Section: Chapter Questions

Problem 3.9P

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