Utility Maximisation and Non-negativity Constraints. Suppose the consumer wants to maximise utility U(C₁, C₂) subject to the budget constraint P₁C₁ + P₂C₂ = m (14) where c₁ is the consumption of good one, c₂ is the consumption of good two and the prices of goods one and two are p₁ and p₂ respectively. Let us assume that P₂ = 1. a. U (C₁, C₂) = ln(c₁) + In (c₂). (15) Without imposing non-negativity constraints on consumption and assuming P2 = 1, calculate the optimal levels of c₁ and c₂ for the consumer. Now suppose, for the remainder of this question, that the utility function is instead given by the following: U(C₁, C₂) = ln (c₁) + ln(1 + C₂). b. What are the marginal utilities of goods one and two, au au ac₁ and ac₂ as c₁ and c₂ approach zero respectively? Explain which should be interpreted as a "luxury good" and why. (16) (17)

Utility Maximisation and Non-negativity Constraints. Suppose the consumer wants to maximise utility U(C₁, C₂) subject to the budget constraint P₁C₁ + P₂C₂ = m (14) where c₁ is the consumption of good one, c₂ is the consumption of good two and the prices of goods one and two are p₁ and p₂ respectively. Let us assume that P₂ = 1. a. U (C₁, C₂) = ln(c₁) + In (c₂). (15) Without imposing non-negativity constraints on consumption and assuming P2 = 1, calculate the optimal levels of c₁ and c₂ for the consumer. Now suppose, for the remainder of this question, that the utility function is instead given by the following: U(C₁, C₂) = ln (c₁) + ln(1 + C₂). b. What are the marginal utilities of goods one and two, au au ac₁ and ac₂ as c₁ and c₂ approach zero respectively? Explain which should be interpreted as a "luxury good" and why. (16) (17)

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

See similar textbooks

Related questions

Q: Q2. Suppose a consumer seeks to maximize the utility function U (x, y) = (x + 2) (y + 1), where and…

A: U(x,y) = (x+2)(y+1) Budget Constraint = Px*x + Py*y = I

Q: 3. A consumer's utility function is given by U(x, y) = min{x,y}. Initially px prices are pz = 1 and…

A: Compensating variation measures the change in income or wealth that would be necessary to keep an…

Q: Suppose a consumer’s preferences can be represented by the utility function U(X,Y) = Min (2X,Y).…

A: Utility function: U = Min (2X, Y) --------------------- Budget constraint: Price of good X (Px) = $2…

Q: A consumer with the following utility function U = xy2 faces two constraints. The first is the…

A: Consumer theory is important in economics because it investigates how consumers might maximize…

Q: Consider a consumer who consumes pizzas and soft drinks. The utility function is given by U (p, s) =…

A: The satisfaction a customer experiences from consuming a combination of services and items is…

Q: Which of the following utility functions represent preferences which do not satisfy monotonicity? U=…

A: Monotonic preferences refer to such consumer preferences towards the greater consumption of goods…

Q: Suppose a consumer has utility function U(X, Y) = X2/3Y1/3. Suppose the consumer has M = $90 to…

A: Utility function : U = X2/3 Y1/3Income : M = 90 Price of X : Px = 15 Price of Y :Py = 6

Q: Milo spends £27 on stamps (S) and maple candy (C). His preferences for these goods are given by the…

A: Given , Milo income = 27 Utility function : U = min { S , 3C } Price of Stamps = 1.1 Price of…

Q: a)Assume that the typical consumer always spends a small share of her overall budget on Vietnamese…

A: Total customers = 120,000 Price of other products = Pm= $1 Price of Vietnamese Meals = Pv The…

Q: Do not use Ai

Q: Utility Function: U(X,Y)=X1/2Y1/2 Budget Constraint: 2X+2Y=16 If the price of good X changes…

A: Utility function: ..........(1)Budget constraint: ......…

Q: uppose a consumer’s preferences for goods Y and X can be represented by the utility function,…

A: A rational consumer will consume where the MUx/MUy = Px/Py Here , MUx is the Marginal utility's of…

Q: Consider the following utility function and budget constraint: U = (beer^.4)*(pizza ^.6) 50 =…

A: The utility function is given as The budget equation is given as

Q: A consumer's utility function is -a u = x1x², where ₁ and 2 are the quantities consumed of goods one…

A: The total level of satisfaction that an individual derives from the consumption of good is referred…

Q: For the utility function U = Qx0.28Q (1-0.28) and the budget 137 = 110x + 6Qy find the CHANGE in…

A: Utility: The purpose of the utility is to satisfy the power of a commodity. It is possible to say it…

Q: find the CHANGE in optimal consumption of X if the price of X increases by a factor of 1.5.

A: Utility is the total satisfaction that the consumer derives from the consumption of good.The…

Q: Jasper utility function for beer and pizza can be expressed as: U(B,P) = BXP Suppose beer costs $5…

A: A utility function is a mathematical representation that quantifies a consumer's preferences or…

Q: Max chooses to purchase movie tickets and restaurant meals every week with his $100. If the price of…

A: A customer experiences a budget constraint when their ability to spend depends on their income. We…

Q: Define the Marginal Rate of Substitution between two goods (X and Y). If a consumer’s preferences…

A: Answer: Definition of the marginal rate of substitution (MRS): The marginal rate of substitution…

Concept explainers

Equilibrium Price and Quantity

Equilibrium refers to a state of balance that comes about because there is equality in all forces.

Question

4. Utility Maximisation and Non-negativity Constraints. Suppose the consumer
wants to maximise utility U(C₁, C₂) subject to the budget constraint
P1C₁ + P₂C₂ = m
(14)
where c₁ is the consumption of good one, c₂ is the consumption of good two and
the prices of goods one and two are p₁ and p2 respectively. Let us assume that
P₂ = 1.
a.
U(C₁, C₂) = ln(c₁) + In (c₂).
(15)
Without imposing non-negativity constraints on consumption and assuming
P₂ = 1, calculate the optimal levels of c₁ and c₂ for the consumer.
Now suppose, for the remainder of this question, that the utility function is
instead given by the following:
U(C₁, C₂) = ln(c₁) + ln(1 + c₂).
b.
What are the marginal utilities of goods one and two,
au
and
au
Әс1
əcz
as c₁ and c₂ approach zero respectively? Explain which should be
interpreted as a "luxury good" and why.
(16)
(17)

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1: Introduction

Step 2: Solution(a)

Step 3: Solution(b)

Solution

Step by step

Solved in 4 steps with 4 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, economics and related others by exploring similar questions and additional content below.

Similar questions

Q2. Suppose a consumer seeks to maximize the utility function U (x, y) = (x + 2) (y + 1), where and y represent the quantities of the two goods consumed. The prices of the two goods and the consumer's income are pa, py, and I. Write out the consumer's budget constraint and the Lagrangian function for the problem.
A consumer has a budget of £12 to split between two goods: good 1 has a price of 2, good 2 has a price of 3. Write the consumer’s budget constraint algebraically. Convert this budget constraint into the formula for the budget line. Show this line on a suitably labelled graph. The consumer’s budget increases to £24. Show the effect of this change graphically. A consumer dislikes good 1, and dislikes good 2. Show these preferences on a suitably labelled graph with an indifference curve. Label the graph: which areas would be preferred to those on the line?
A consumer consumption bundle contains good X and Y. Suppose the price of good X goes up and a consumer goes on consuming the exact same amount of X as before. Check the correct statement(s).A) Both X and Y are inferior. B) X is an inferior good.C) In the special case where X is Leisure, and Y is "all other goods", X is an inferior good.D) The consumption of Y decreases.E) It is impossible that the consumption of X remains unchanged when the price of × goes up.F) The substitution effect is negative
Assume, as in Exercise 22.1, that a consumer has utility function F or fruit and chocolate. Determine the consumer's demand functions q1(P1, P2, M) and q2(P1, P2, M). Determine also It* in terms of P1, P2 and M. Find the indirect utility function and show that It* = 8Vj8M. Suppose, as before, that fruit costs $1 per unit and chocolate $2 per unit. If the income is raised from $36 to $36.5, determine the precise value of the resulting change in the indirect utility function. Show that this is approximately equal to (O.5)λ*, where λ* is evaluated at P1 = 1,P2 = 2 and M = 36. Exercise 22.1 A consumer purchases quantities of two commodities, fruit and chocolate, each month. The consumer's utility function is For a bundle (X1, X2) of X1 units of fruit and X2 units of chocolate. The consumer has a total of $49 to spend on fruit and chocolate each month. Fruit cost $1 per unit and chocolate costs $2 per unit. How many units of each should the consumer buy…
A consumer’s utility only depends on the consumption of goods A and B according to the following Cobb-Douglass utility function: U(A, B) = A1/4 B 3/4. The price of goods A and B are $20 and $40, respectively. The consumer has a budget of $1200 that he can use to consume the two goods. a. Write down the budget constraint and plot it. b. Calculate the optimal bundle and maximized utility for the consumer. c. A new tax of $10 is imposed on the price of good B. Compute the new optimal bundle of good A and B for the same consumer. What is the utility loss due to the tax? d. Show that the consumer would prefer a lump sum income tax that raises the same revenue as the tax on good B. Note:- Do not provide handwritten solution. Maintain accuracy and quality in your answer. Take care of plagiarism. Answer completely. You will get up vote for sure.
Alice receives an allowance of 500 dollars that she spends on buying snacks (S) and tea (T). The price of each snack is 10 dollars and the price of each tea is 5 dollars. Her utility is given by: U (S, T) = 2S³/4 +T3/4 (a) Find her marginal rate of substitution (MRS) between S and T. (b) Write Alice's budget constraint. (c) Find Alice's optimal consumption and the optimal A. (d) What is her new consumption if the price of tea becomes 10 dollars? Note: numeric solutions for questions (c) and (d) are not integers.
Suppose Jeff has a perfect substitute preferences for watching movies (M) and TV (T) shows. Jeff's MRSTM is such that he is always indifferent between watching one movie and five TV shows independent of the amount of goods consumed. Which of the following statements is true? If Jeff had a choice between 1 movie and 1 TV show before bed, he would choose to watch the movie. If Jeff has already watched 3 movies and no TV shows, then he would prefer to watch a TV show next. He is indifferent between watching 1 TV show or 1 movie before bed. Jeff would prefer watching 2 TV shows and no movies over watching 2 Movies and no TV shows.
Consider a consumer with M = $120 to spend (income) and faces prices PX = $15 and PY =$5 for goods X and Y. Her utility function is U(X, Y) = X1/2 + Y1/2.a Carefully. express the consumer’s choice problem, using the given information (this is whereyou write out the max operator, the choice variables, the objective function, and the budget constraint). b. Compute the absolute value of the consumer’s marginal rate of substitution, and inspect it todetermine the shape of the consumer’s indifference curves: C-shaped, linear, )-shaped, or some other shape.To show your work, neatly use the arrow argument, increasing X (↑) and decreasing Y (↓) to see whether|MRS(X, Y)| is diminishing along an indifference curve. c. If the indifference curves are C-shaped write out the budget line and the equal slopes conditionthat characterize an interior solution to the consumer’s choice problem. Use the particulars for the givenconsumer. Solve these conditions to find the interior solution. On the other…
There are two goods, x and y. The consumers' utility is U=4x6y³. The price of x is Px=5, the price of y is Px=10. At interior optimum, the relationship between x and y is: options: y=2x y=0.25x y=0.5x y=x
6. Consider a consumer with the utility function u(x,, x2) = In(x,) + x2 and the budget constraint p, x, + PP2X2 = m. Derive the consumer's demand functions for x, and x2.