part F G H 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. Examine how the equilibrium price p changes with respect tothe endowment of good 1, i.e. with respect to w for i = 1,2, and with re-spect to the endowment of good 2, i.e. with respect to w for i = 1,2. Give(d)intuition for these results.(HINT: You may, but need not, take respective derivatives of the expres-sion for equilibrium price with respect to each variable of interest to estab-lish the comparative statics asked for here. Simple observation will sufficehere.)Examine how the equilibrium price p changes with respect to(e)the taste for good 1, that is, with respect to a and B. Give intuition for theseresults. (HINT: Same remark as for the previous part)(f)each good is 4, and that w = 1, w =3, } =3, w} = 1. Assume further thata = B = 1/2. Plug these value into the expressions found above to find theequilibrium price and allocations.For the following parts, assume that the total endowment for(3)economy with these values.Define what is meant by a core, and determine the core in this(h)Draw an Edgeworth box, taking care to reflect the initial en-dowments, the Pareto frontier, the contract curve, the core and the equi-librium allocation this economy. You need not find an exact solution forthe indifference curves, but only a graphical representation.

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, x3) = (x}P(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. For each consumer i = 1,2, use the first-order conditions to (b) 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 p; is given by the expression aw;+Bw? [(1-a)w} +(1-B)w}] Define Walras' here and show that this holds here.

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, x3) = (x}P(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. For each consumer i = 1,2, use the first-order conditions to (b) 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 p; is given by the expression aw;+Bw? [(1-a)w} +(1-B)w}] Define Walras' here and show that this holds here.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter4: Utility Maximization And Choice

Section: Chapter Questions

Problem 4.14P

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