1. Suppose there are two consumers, A and B. The utility functions of each consumer are given by: UA(X,Y) = X*Y UB(X,Y) = X*Y3 Therefore: • For consumer A: MUX = Y; MUY = X • For consumer B: MUX = Y3; MUY = 3XY2 The initial endowments are: A: X = 10; Y = 6 B: X = 14; Y = 19 show all work a) Suppose the price PY = 1. Calculate the price of X, PX that will lead to a competitive equilibrium. b) How much of each good does each consumer demand in equilibrium? Consumer A’s Demand for X: Consumer A’s Demand for Y: Consumer B’s demand for X: Consumer B’s demand for Y: c)What is the marginal rate of substitution for consumer A at the competitive equilibrium?

1. Suppose there are two consumers, A and B. The utility functions of each consumer are given by: UA(X,Y) = XY UB(X,Y) = XY3 Therefore: • For consumer A: MUX = Y; MUY = X • For consumer B: MUX = Y3; MUY = 3XY2 The initial endowments are: A: X = 10; Y = 6 B: X = 14; Y = 19 show all work a) Suppose the price PY = 1. Calculate the price of X, PX that will lead to a competitive equilibrium. b) How much of each good does each consumer demand in equilibrium? Consumer A’s Demand for X: Consumer A’s Demand for Y: Consumer B’s demand for X: Consumer B’s demand for Y: c)What is the marginal rate of substitution for consumer A at the competitive equilibrium?

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter4: Utility Maximization And Choice

Section: Chapter Questions

Problem 4.12P

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Exercise 4 Consider an economy with two consumers, Alexia and Bart, who live two periods, t = 0 and t = 1. In each period they can consume one type of good and their preferences for consumption are given by U (co, ci) = c(ci)² _i = A, B. Alexia and Bart have the following endowment of good in each period M=1, M₁ = 1, MB = 2, MB = 2. In t = 0, Alexia and Bart can exchange a financial contract for the delivery of one unit of consumption good in t = 1 (a bond). Name p the price of the bond and b² the amount bought by agent i = =A, B. (a) Write down each agent's utility maximization and budget constraints assuming that he/she can trade the bond without restrictions. (b) Find each agent's optimal quantity b² as a function of the bond net return r. (c) Find the equilibrium value of r and the equilibrium demand/supply of each agent.
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Suppose there are two consumers. A and B. The utility functions of each consumer are given by: U₁XX)-x²-y U₂XY)-x-² Therefore For consumer A: MUX-2XY: MU-X² For consumer B: MUx-Y²; MUy-2XY The initial endowments are: A:X-60; Y-150 B.X-45: Y-75 al Suppose the price of Y, Py-1. Calculate the price of X, P, that will lead to a competitive equilibrium. How much of each good does each consumer demand in equilibrium? Consumer A's demand for X: Consumer A's demand for Y Consumer B's demand for X: Consumer B's demand for Y d ts) What is the marginal rate of substitution for consumer A at the competitive equilibrium?
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Suppose Jimi has reference dependent preferences over guitars and money as in Tversky and Kahneman (1991). His utility functions are given below. Gains Gains 400 -2 -2 2 Guitars 2$ Losses Losses i-600 -2 What is the least amount of money Jimi is willing to accept to sell one of his guitars? (just enter a dollar amount, i.e., "10o0", not "$1000"
2. The standard model of consumer behavior assumes that income is exogenous. Of course, a person's income usually depends upon the number of hours the individual works. Suppose that an individual has T hours each day which can be allocated toward working time, H, or leisure time, L-that is, T = H + L. The individual earns w dollars for each hour worked. Then the individual's income is M + wH where M denotes any non-labor income. The individual has preferences over leisure time, L, and a consumption good, X, which can be represented by a quasi-concave utility function, U(X, L). The consumption good, X, can be purchased at the price Px. The individual seeks to maximize the utility subject to the constraints on time and money. (1) Formulate the individual's problem as an optimization problem with two constraints a time constraint and a money constraint. Derive the first-order conditions. Give an economic interpretation of the Lagrange multipliers. (2) The two-constraints problem can be…
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