Anna and Bob are the only residents of a small town. The town currently funds its fire department solely from the individual contributions of these two residents. Each of the two residents has a utility function over private goods a and total number of firemen M, of the form: u(x, M) = 2 lnx+ In M. The total provision of firemen hired, M, is the sum of the number hired by each of the two persons: M = MA + MB.Ann and Bob both have income of 200 each, and the price of both the private good and a fireman is 1. They are limited to providing between 0 and 200 firemen. For the purposes of this problem, you can treat the number of firemen as a continuous variable (it could

Anna and Bob are the only residents of a small town. The town currently funds its fire department solely from the individual contributions of these two residents. Each of the two residents has a utility function over private goods a and total number of firemen M, of the form: u(x, M) = 2 lnx+ In M. The total provision of firemen hired, M, is the sum of the number hired by each of the two persons: M = MA + MB.Ann and Bob both have income of 200 each, and the price of both the private good and a fireman is 1. They are limited to providing between 0 and 200 firemen. For the purposes of this problem, you can treat the number of firemen as a continuous variable (it could

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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Anna and Bob are the only residents of a small town. The town currently funds its fire department solely from the individual contributions of these two residents. Each of the two residents has a utility function over private goods xx and total number of firemen MM, of the form: u(x,M)=2lnx+lnMu(x,M)=2ln⁡x+ln⁡M. The total provision of firemen hired, MM, is the sum of the number hired by each of the two persons: M=MA+MBM=MA+MB. Ann and Bob both have income of 200 each, and the price of both the private good and a fireman is 1. They are limited to providing between 00 and 200200 firemen. For the purposes of this problem, you can treat the number of firemen as a continuous variable (it could be man-years). How many firemen are hired if the government does not intervene?
Anna and Bob are the only residents of a small town. The town currently funds its fire department solely from the individual contributions of these two residents. Each of the two residents has a utility function over private goods x and total number of firemen M, of the form: u(x,M)=2ln⁡x+ln⁡M. The total provision of firemen hired, M, is the sum of the number hired by each of the two persons: M=MA+MB. Ann and Bob both have income of 200 each, and the price of both the private good and a fireman is 1. They are limited to providing between 0 and 200 firemen. For the purposes of this problem, you can treat the number of firemen as a continuous variable (it could be man-years). Consider the setup from above. Suppose that the government recruits additional ?N firemen and taxes Ann and Bob equally to cover the cost. Therefore, the total number of firemen is MA+MB+N, where MA,MB are appropriate individually-optimal contributions of A and B(i.e., the agents behave optimally, conditional on the…
Anna and Bob are the only residents of a small town. The town currently funds its fire department solely from the individual contributions of these two residents. Each of the two residents has a utility function over private goods x and total number of firemen M, of the form: u(x,M)=2ln⁡x+ln⁡M. The total provision of firemen hired, M, is the sum of the number hired by each of the two persons: M=MA+MB. Ann and Bob both have income of 200 each, and the price of both the private good and a fireman is 1. They are limited to providing between 0 and 200 firemen. For the purposes of this problem, you can treat the number of firemen as a continuous variable (it could be man-years). Consider the setup from above. The government proposes an alternative, market-based solution. They charge each citizen the price p for every firemen stationed at the local fire station. Then, the price is being set at a level p∗ at which each individual demands the socially optimal number of firemen. What is the…
Anna and Bob are the only residents of a small town. The town currently funds its fire department solely from the individual contributions of these two residents. Each of the two residents has a utility function over private goods x and total number of firemen M, of the form: u(x,M)=2ln⁡x+ln⁡M. The total provision of firemen hired, M, is the sum of the number hired by each of the two persons: M=MA+MB. Ann and Bob both have income of 200 each, and the price of both the private good and a fireman is 1. They are limited to providing between 0 and 200 firemen. For the purposes of this problem, you can treat the number of firemen as a continuous variable (it could be man-years). Consider the setup from above. Suppose that we identify the social welfare in this town by the sum of utilities of Ann and Bob. What is the socially optimal number of firemen? a. M=400/3 b. M=80 c. M=400 d. m=200/3
Anna and Bob are the only residents of a small town. The town currently funds its fire department solely from the individual contributions of these two residents. Each of the two residents has a utility function over private goods ? and total number of firemen ?, of the form: ?(?,?)=2ln?+ln?. The total provision of firemen hired, ?, is the sum of the number hired by each of the two persons: ?=??+??. Ann and Bob both have income of 200 each, and the price of both the private good and a fireman is 1. They are limited to providing between 0 and 200 firemen. For the purposes of this problem, you can treat the number of firemen as a continuous variable (it could be man-years). Consider the setup from Question 2. Suppose that the government recruits additional ?N firemen and taxes Ann and Bob equally to cover the cost. Therefore, the total number of firemen is ??+??+?MA+MB+N, where ??,??MA,MB are appropriate individually-optimal contributions of ?A and ?B(i.e., the agents behave optimally,…
1). Anna and Bob are the only residents of a small town. The town currently funds its fire department solely from the individual contributions of these two residents. Each of the two residents has a utility function over private goods ?x and total number of firemen ?, of the form: ?(?,?)=2ln?+ln?. The total provision of firemen hired, ?, is the sum of the number hired by each of the two persons: ?=??+??. Ann and Bob both have income of 200 each, and the price of both the private good and a fireman is 1. They are limited to providing between 0 and 200 firemen. For the purposes of this problem, you can treat the number of firemen as a continuous variable (it could be man-years). How many firemen are hired if the government does not intervene? a. ?=80 b. ?=400/3 c. ?=200/3 d. ?=0 Consider the setup from Question 1. Suppose that we identify the social welfare in this town by the sum of utilities of Ann and Bob. What is the socially optimal number of firemen? a. ?=400/3 b.…
Consider a couple whose behaviour follows the unitary household model. Their preferences can be represented by the utility function: U(CM; CH) = min (CM, CH), where CM, denotes market goods and CH denotes home production. Each spouse can work up to 50 hours per week, and those 50 hours can be divided between market work and home production. Joe and Anna are each paid £20 per hour for market work. Joe produces £20 of home production per hour, while Anna produces £30 per hour of home production. (a)How many hours are each of the spouses allocating to home production and market work? and Suppose that Anna is offered a pay raise, so that her hourly market wage increases to £25, and nothing else changes. Will that change the identity of the spouse who works more hours on the market? Explain your answer.
3. A firm that is located in country H, where price levels are p = (1,1), needs to send one of its two employees to its branch in country F. However, in country F price levels are p′ ≫ p, so the firm will have to pay additional salary to ensure that its employee is equally well-off in country F as she was in country H. Suppose the utility functions of the two employees are u1(x1, x2) = x1 + x2 and u2(x1, x2) = min {x1, x2}. The two employees are otherwise identical, including current salary. If the firm wants to minimize the additional salary it needs to pay, which employee should it send? Explain.
Billy John Pigskin of Mule Shoe, Texas, has a von Neumann-Morgenstern utility function of the form u(c) = √c. Billy John also weighs about 300 pounds and can outrun jackrabbits and pizza delivery trucks. Billy John is beginning his senior year of college football. If he is not seriously injured, he will receive a $1,000,000 contract for playing professional football. If an injury ends his football career, he will receive a $10,000 contract as a refuse removal facilitator in his home town. There is a 10% chance that Billy John will be injured badly enough to end his career. If Billy John pays $p for an insurance policy that would give him $1,000,000 if he suffered a career-ending injury while in college, then he would be sure to have an income of $1,000,000 − p no matter what happened to him. Write an equation that can be solved to find the largest price that Billy John would be willing to pay for such an insurance policy. Here is my question: Why is Billy's income 1,000,000 - p even…
A husband and wife would produce incomes Yh and Yw in their fallback situations. The utility each derives in any circumstance is just equal to his or her consumption expenditure in that circumstance. In their fallback situations, their consumption expenditure levels are just equal to their incomes. Thus their fallback levels of utility are Yh and Yw. If they cooperate, they produce Z>Yh + Yw. They engage in Nash cooperative bargaining to determine how to allocate Z across the consumption of the husband, Ch, and consumption of the wife, Cw, subject to the budget constraint that Ch + Cw = Z. Under any bargained allocation, the two would derive utilities of Ch and Cw. a) The surplus associated with cooperation is S = Z − Yh − Yw. Show that each spouse consumes his or her fallback income plus half the surplus in the Nash cooperative bargaining solution. Please do fast ASAP fast please.
A husband and wife would produce incomes Yh and Yw in their fallback situations. The utility each derives in any circumstance is just equal to his or her consumption expenditure in that circumstance. In their fallback situations, their consumption expenditure levels are just equal to their incomes. Thus their fallback levels of utility are Yh and Yw. If they cooperate, they produce Z>Yh + Yw. They engage in Nash cooperative bargaining to determine how to allocate Z across the consumption of the husband, Ch, and consumption of the wife, Cw, subject to the budget constraint that Ch + Cw = Z. Under any bargained allocation, the two would derive utilities of Ch and Cw. What do Ch and Cw equal if Yh = Yw (but this quantity is not equal to zero)? Please do fast ASAP fast
a) Consider a couple whose behaviour follows the unitary household model. Their preferences can be represented by the utility function: U(CM; CH) = min (CM, CH), where CM, denotes market goods and CH denotes home production. Each spouse can work up to 50 hours per week, and those 50 hours can be divided between market work and home production. Joe and Anna are each paid £20 per hour for market work. Joe produces £20 of home production per hour, while Anna produces £30 per hour of home production. How many hours are each of the spouses allocating to home production and market work? b) Suppose that Anna is offered a pay raise, so that her hourly market wage increases to £25, and nothing else changes. Will that change the identity of the spouse who works more hours on the market? Explain your answer. [Hint there is no need to calculate the full solution to this case]