A. Consider a consumer whose preferences can be represented by Cobb-Douglas utilityfunction u(x₁, x₂) = xx where ₁ and 2 are the quantities of good 1 and good 2she consumes. Let p₁ and p2 be the prices of good 1 and good 2 and let m denote herincome.1. Derive the consumer's Marshallian demand functions.2. Derive the consumer's Hicksian demand functions.3. Derive the consumer's expenditure function.4. Let m = 20, P₁ = 2, and p2 = 1. Suppose that the price of good 1 drops to p₁ = 1.Find the following(a) Compensating variation (CV)(b) Equivalent variation (EV)(c) Change in consumer surplus (ACS)(d) Compare CV, ACS, and EV.5. Let m = 120, P₁ = 1, and p2 = 1. Suppose that the price of good 1 increases toP₁ = 2. Find the following(a) Compensating variation (CV)(b) Equivalent variation (EV)(c) Change in consumer surplus (ACS)(d) Compare CV, ACS, and EV. B. Albert works for a construction company. He has 70 hours a week available todivide between construction work and leisure, and he has no other sources of income.Albert is allowed to work as many hours (not exceeding 70) as he wishes to. His utilityfunction is U(c, r) = cr where c is his income spent on goods and r is the number ofhours of leisure that he has per week.• Suppose that the hourly wage is w dollars. Find Albert's demand for leisure as afunction of w. How many hours will Albert work at wage w?1• Albert is currently being paid $5 per hour. How many hours will he choose towork?• The company considers to raise Albert's hourly wage from $5 to $10. How manyhours will he choose to work at the new wage?• Assume now that the company offer "double pay" for overtime. That is, Albert ispaid $10 an hour for every hour beyond the amount he chose in part 2. Draw hisnew budget set and find the number of hours he will work under this new wagescheme.. Find the substitution and income effects on leisure if the hourly wage changes from$5 to $10.

Given Consumer utility function: u(x1,x2)=x113x223 .......(1) The price of good 1…

= A. Consider a consumer whose preferences can be represented by Cobb-Douglas utility function u(x1, x₂) = xx where ₁ and 2 are the quantities of good 1 and good 2 she consumes. Let p₁ and p2 be the prices of good 1 and good 2 and let m denote her income. 1. Derive the consumer's Marshallian demand functions. 2. Derive the consumer's Hicksian demand functions. 3. Derive the consumer's expenditure function.

= A. Consider a consumer whose preferences can be represented by Cobb-Douglas utility function u(x1, x₂) = xx where ₁ and 2 are the quantities of good 1 and good 2 she consumes. Let p₁ and p2 be the prices of good 1 and good 2 and let m denote her income. 1. Derive the consumer's Marshallian demand functions. 2. Derive the consumer's Hicksian demand functions. 3. Derive the consumer's expenditure function.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter6: Demand Relationships Among Goods

Section: Chapter Questions

Problem 6.9P

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A. Consider a consumer whose preferences can be represented by Cobb-Douglas utility function u(x₁, x2) = x³x² where x₁ and ⁄2 are the quantities of good 1 and good 2 she consumes. Let p₁ and p2 be the prices of good 1 and good 2 and let m denote her income. Derive the consumer's Marshallian demand functions. Derive the consumer's Hicksian demand functions. Derive the consumer's expenditure function. Let m = = 120, p₁ = 2, and p2 = 1. Suppose that the price of good 1 drops to p₁ = 1. Find the following Compensating variation (CV) Equivalent variation (EV) Change in consumer surplus (ACS) Compare CV, ACS, and EV. Let m = 120, P₁ P₁ = 2. Find the following = 1, and p2 = 1. Suppose that the price of good 1 increases to Compensating variation (CV) Equivalent variation (EV) Change in consumer surplus (ACS) Compare CV, ACS, and EV.
a) Assume that an individual consumes two goods, and achieves the benetby çonsuming respectively 1 x and x units of each of the goods. Define the terms indifference curve and budget condition, and show the consumer benefit-maximizing good combination in a chart. Explain why this one the combination solves the utility maximization problem. b) Assume that the preferences of the consumer can be expressed by the utility function W),4 + X, 2 and that the price per unit x 1 is p =, the price per unit x 2 is 2 p= and that the income ( m ) is 200. Find the consumer's optimal choice in this case.
Suppose a consumer purchases food X and clothing Y. The utility function of the consumeris given by: U(X,Y ) = XY + 5Y and the budget /income/ of the consumer is 200 ETB. Andalso the price of food is X P and the price of clothing is Y P . Then drive the equation ofconsumer’s demand function for food and clothing.
1. Kofi consumes two goods (x) and (y). The following utility function represents his preferences: u(x, y) = 0.4 ln x+0.6 ln y Br Paxt by Y Suppose the prices of the two goods (x and y) are p, and p, and his budget is M. You are required to do the following: a) Specify Kofi's budget set. - b) Derive the Marshallian demand function for each of the two commodities c) Derive the indirect utility function and verify the Roy's identity. d) How many units of each of the two commodities will be consumed by Kofi if his income is GHS100.00 and p. = 1; p, = 2?
A consumer has the following utility function: Ulx, y) = xy -y, *21 where x and y represents the quantities consumed of goods X and Y. y 20 What will be the substitution and income effects for X and Yassuming that the consumer attempts to maintain the same level of utility achieved before price of Y increased (that is, when price of Y was $1)? SEx= +0.5 IEx = -0.5 SE, = -0.25 IE- = -0.25 SEx= +0.293 IE = -0.293 SEy = -0.414 IE, = +0.414 SEr= +0.25 IE SE, = -0.75 IE, = -0.75 = -0.25 SEx= +0.414 IEx = -0.414 SEy = -0.293 IE, = -0.207 Income = $3 Px= $1, Py= $2
Let the following table represents the total utility of a given consumer, in the cardinal utility approach A) Calculate the MUx and MUy and fill the table in the 4th and 5th rows. B) If the two products (X&Y) are free goods how many of X and Y should the cons consumer take to maximize utility? C) What is the maximum utility of X and Y if they are free?. D) Assuming the consumer has any amount of money (enough budget) how many of X and Y should the consumer buy, to maximize utility? E) What is the total utility of X and Y? F) Let now price of X is 4 birr per unit and price of Y is 2 birr per unit and budget of the consumer for consumption of X and Y is 20 birr. Given budget constraint how many of X and Y should the consumer buy to maximize utility? G) What are the total utility of X and Y
2. Consider these three utility functions: u1 = U2 = In + In a2 Which of these functions can be said to represent the same preferences: Demonstrate whether uz and uz are monotonic transformations of the first function, u1.
Suppose that the consumer maximizes his utilitysuch that the marginal utilities of goods X and Y are; MUx=80-16x MUy=60-2y Let Px=P4,Py=P2 and I=P80 Find the utility maximizing quantities X and Y
5. Consider a consumer that seeks to minimize his expenditure E to achieve a given 1/41/4 level of utilityŪ. Assume that E = p₁x₁ + p₂x²; Ū=x^x^; and p₁ and p₂ are given. a) Set up the Lagrangian. b) Show the first-order conditions or minimization. c) Derive the expressions for the optimal levels of x, and x₂. d) Using the second-order conditions, verify if the solution generates a minimum value for E (Use H₂ to verify).
1 Consider a consumer who consumes only two goods, 21 and z2. His utility function is u (21, 22) = In z1 + 2. where In represents the natural logarithm. 1.a Derive the consumer's MRS of good 1 for good 2 using calculus to calculate his marginal utility from z and his marginal utility from z2. 1.b If the price of z1 is $2 per unit and the price of z2 is $4 per unit, and the consumer's income is $100, What is the equation of this consumer's budget line? 1.c What are the optimal consumption choices, z and z5, for this consumer? Show your work.
2. Ping receives a weekly allowance to purchase soda (x) and juice (y) at school. His preferences over these two goods can be represented by the utility function U(x,y) =0.5x+ 5lny where x represents the number of cans of soda and y represents the number of juice boxes. a) Given his preferences find his Marshallian demand functions for soda (x) and juice (y). b) Suppose that the price of a can of soda is $1.5 and that Ping has $30 to spend on soda and juice. Write Ping's demand curve for juice (y). Illustrate his demand curve. c) Suppose that the price of a box of juice is $1 (the price of soda and income remain $1.5 and $30, resp.). Use your demand functions to find his best bundle. In an indifference curve diagram illustrate his best bundle at these prices. For the remainder of the question assume that Ping's allowance (income) rises to $36 per week and the prices of the two goods are unchanged at Px= $1.5 and Py= $1. d) Use your demand functions to find his new best bundle. e)…
A consumer’s preferences between goods x and y are representedby the utility function u(x, y) = 2min{x, y}+10. Suppose this consumer hasincome of $16, the price of good x is $3 and the price of good y is $1. Suppose the price of good x increases to $7 while the price of good y andthe consumer’s income stay constant. Calculate the magnitudes of the compensating and the equivalent variations. Explain what each measures.