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.
Chapter1: Making Economics Decisions
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![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})!-«, while
Consumer 2 has utility function u(x, x;) = (xP(x)-P. 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.
Consider an exchange economy with two
Write 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 to
determine the demand functions for each consumer i = 1,2 and for each
good j = 1,2, in terms of the price p.
(c)
Find the aggregate demand for each good j = 1,2 and clear the
markets for each good. Hence, show that the equilibrium price pi is given
by the expression
aw; +Bw;
[(1-a)w} +(1-B)w}]
Define Walras' here and show that this holds here.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F28a22340-6d9d-4cc4-a9c4-adb517e05df3%2F08004fec-1148-4f49-b32c-4c4b0d0c41e7%2F6ozd6hk_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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})!-«, while
Consumer 2 has utility function u(x, x;) = (xP(x)-P. 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.
Consider an exchange economy with two
Write 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 to
determine the demand functions for each consumer i = 1,2 and for each
good j = 1,2, in terms of the price p.
(c)
Find the aggregate demand for each good j = 1,2 and clear the
markets for each good. Hence, show that the equilibrium price pi is given
by the expression
aw; +Bw;
[(1-a)w} +(1-B)w}]
Define Walras' here and show that this holds here.
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