Re: Daily question

Posted by Amit Goyal on Mar 10, 2009; 3:59am
URL: http://discussion-forum.276.s1.nabble.com/Daily-question-tp2328964p2453206.html

8-12 March, 2009

A cab was involved in a hit and run accident at night. Two cab companies, the green and the blue, operate in the city. You are given the following data:
a) 85% of the cabs in the city are green and 15% are blue.
b) A witness identified the cab as blue. The court tested the reliability of witness under the same circumstances that existed on the night of the accident and concluded that the witness correctly identified each one of the two colors 80% of the time and failed 20% of the time.
What is the probability that the cab involved in the accident was blue rather than green?

At each generation a microbe either splits into two perfect copies of itself or dies. If the probability of splitting is p, what is the probability that a single microbe will produce an everlasting colony?

Consider the following situation where the consumer's utility is given by u=min{x,y}.
Suppose px=py=1/2, M=1.
1) What will be the optimal consumption bundle of the utility maximizing individual?
2) Now suppose for promoting good x, company announces buy one get one free. Will that change the optimal consumption bundle of the individual in question? Will his utility increase at the new consumption bundle?
3) Now suppose individual just cares about integer amounts of good x and good y consumed in optimum and does not care about fractions. In other words his utility is given by u=min{[x],[y]} + (I/2) where I is the income left after spending on x and y.
Compare the two situations:
a) Company has an offer of buy one get one free.
b) Company offers 50% off on the purchase of good x i.e. effective price of good x is 1/4.
Suppose Marginal cost of producing good x is 1/8. Which of the above two situations is better for consumer and which one is better for the producer? Compare the profits and utility of the consumer and producer in above two cases.
[-] is the greatest integer function i.e [x] is the greatest integer less than or equal to x.
Eg--> [1.4]=1, [0.3]=0, [2]=2, [-1.3]=-2

Consider the following situation in which there is a need for designing a rent sharing mechanism for an apartment:
Assume its a two bedroom apartment and there are two individuals(1, 2). One room is bigger than the other. Lets call the bigger room B and smaller room S. The total rent of the house is Rs.10,000. Assume both individuals have the capacity to pay off the entire rent by themselves i.e. each of them have atleast Rs.10,000 in their pocket. Suppose both Individuals have the same utility function
U(B=1,S=0,R)= 7000-R, where R is the rent paid by the individual and B=1 denotes that he occupied bigger room and not the smaller room.
U(B=0,S=1,R)= 5000-R
U(B=1,S=1,R)= 9000-R
Answer the following:
1) Suggest a rent sharing scheme that ensures equal utility for both the individuals as well as both rooms are occupied. Is the allocation you suggested pareto efficient?
2) Does there exist a pareto efficient allocation in which one individual get both the rooms? If yes, give the allocation. If no, explain why?

There are 6000 commuters who commute to work from Delhi to Gurgaon. There are two ways to make the trip: one is to drive on NH8 and the other on MG road. MG road is uncongested and it takes 45 minutes to commute to work. Travel time on NH8 however, depends on the number of people: if there are N commuters, then it takes (20 + (N/100)) minutes.
(a) If commuters are free to choose their routes, how many will take MG road? How many will take NH8? Calculate the total number of person-minutes per day spent by commuters.
(b) The Delhi Government wants to minimize the total person-minutes spent on commuting. How many commuters should be allocated to NH8 and MG road respectively?