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.

There are two consumers : i = 1 ,2 Two goods : j = 1,2 Utility function for consumer 1 : u1(x11 ,…

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;) = (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 (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. 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. Find the aggregate demand for each good j = 1,2 and clear the (c) markets for each good. Hence, show that the equilibrium price p is given by the expression aw; + Bo [(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, x;) = (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 (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. 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. Find the aggregate demand for each good j = 1,2 and clear the (c) markets for each good. Hence, show that the equilibrium price p is given by the expression aw; + Bo [(1-a)w} +(1-B)w;] Define Walras' here and show that this holds here.

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Person A has a utility function of and person B has a utility function Agent A and agent B have identical endowments of (1/2,1/2). (a) Illustrate this situation in an Edgeworth box diagram. (b) What is the equilibrium relationship between p1and p2? (c) What is the equilibrium allocation?
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Consider an economy with 2 goods and 2 agents. The Örst agent has the utilityfunction, u (x1; x2) = ln x1 + 2 ln x2, and the other one has u (y1; y2) = 2 ln y1 + ln y2.The aggregate endowments of the 2 goods are given by (50; 100). Suppose there is asocial planner who cares about agents equally.(a) Set up the plannerís problem b) Calculate the first-best outcome (i.e., the social plannerís solution).
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. Answer all parts (a) (c) of this question. (a) Consider an agent whose preferences over any couple (x1, x2), where 2₁ ER+ and x2 € R+, e.g., apples and oranges, is such that she prefers the bundle that is closest to having the same number of apples and oranges. Write a utility function u: R² → R+ which represent these preferences. A politician remarks "Our recent increases in the wage rates of teachers has been a total suc- cess! The shortage of teachers has been reduced drastically. Another, similar wage increase should eliminate this shortage entirely" (b) Explain and illustrate in a diagram what is meant by "income effects" and "substitution effects" of a wage rate change. (c) Explain and illustrate how you would model the labour supply decision of a potential teacher. Do you agree that the wage increase will increase the labour supply in this case? Carefully outline the assumptions underlying your argument.
2. General Equilibrium Consider a market with two goods, x and z, and two consumers, A and B. The utility functions for consumers A and B are as follows UA ÜB 2 1 == XBZB and the initial endowments for each consumer are @ eA = (4,2) еB = (2,6) where consumer B is endowed with 2 unit of good x and 6 units of good z, respectively. a) Draw the Edgeworth Box (Don't worry about the shape of the utility curves. Just pick a general shape that we have used before). b) Derive the contract curve.
5.18 In a two-good, two-consumer economy, utility functions are u¹ (x₁, x₂) = x₁(x₂)², u² (x₁, x₂) = (x₁)²x₂. Total endowments are (10, 20). (a) A social planner wants to allocate goods to maximise consumer 1's utility while holding con- sumer 2's utility at u² = 8000/27. Find the assignment of goods to consumers that solves the planner's problem and show that the solution is Pareto efficient. (b) Suppose, instead, that the planner just divides the endowments so that e¹ = (10, 0) and e² = (0, 20) and then lets the consumers transact through perfectly competitive markets. Find the Walrasian equilibrium and show that the WEAs are the same as the solution in part (a).
***Please provide an answer and explanation for BOTH parts (A & B). B builds on and cannot be completed without A.
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