Daily Question solutions

[Amit Goyal]

15 Feb 2009
Pick the correct answer from the options.
a) Indifference curves are downward sloping because:
1) Diminishing Marginal rate of substitution of good x for good y.
2) Marginal Utility from both goods is positive.
3) Marginal utility of one good is positive and other good is negative.
4) Prices of both goods are positive.
Ans. Marginal Utility from both goods is positive.

Do the following problem:
b) Suppose U=xy. Price of x is Rs. 2 and price of y is Rs. 4 and consumer has Rs.600 to spend. Compute the equilibrium bundle. Now suppose price rise to Rs.4. Decompose the total price effect into Substitution effect and Income effect. (Using both Hicksian and Slutsky methods)
Ans. Slutsky: (150,75) ---> (112.5, 112.5)---> (75, 75); Hicks: (150,75) ---> (75√2, 75√2)---> (75, 75)

Monty Hall Game
Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, pigs. You pick a door, say No. 1, and the host, who knows what's behind the other doors, opens another door, say No. 3, which has a pig. He then says to you, 'Do you want to pick door No. 2?' Is it to your advantage to take the switch?
Ans. Yes

Link: http://www.shodor.org/interactivate/activities/SimpleMontyHall/
The activity in the link allows the user to simulate the standard Monty Hall Game.
The user chooses a door, and then another door is opened to reveal a stinking pig. The player must then decide to stick with the original choice or change.

16 Feb 2009
To encourage Elmer's promising tennis career, his father offers him a prize if he wins (atleast) two tennis sets in a row in a three set series to be played with his father and the club champion alternately:
father-champion-father or champion-father-champion, according to Elmer's choice. The champion is a bettter player than Elmer's father. Which series should Elmer choose?
Ans. champion-father-champion

Suppose Asha's preferences between two commodities x and y can be represented by u(x,y)=min{x-5,y+3}. Given an income of Rs.73 and px=3 and py=4. Find Asha's optimal consumption bundle.
Ans. (15, 7)

Feb 17-20, 2009
In answering a question on a multiple choice test a student either knows the answer or he guesses. Let p be the probability that he knows the answer and 1-p the probability that he guesses. Assume that a student who guesses at the answer will be correct with probability 1/m, where m is the number of multiple choice alternatives. What is the conditional probability that a student knew the answer to a question given that he answered it correctly?
Ans. mp/(mp+(1-p))

A miner is trapped in a mine containing three doors. The first door leads to a tunnel which takes him to safety after two hours' travel. The second door leads to a tunnel which returns him to the mine after three hours' travel. The third door leads to a tunnel which returns him to the mine after five hours' travel. Assuming that the miner is at all times equally likely to choose any one of the doors, what is the expected length of time until the miner reaches safety?
Ans.  10

A family has two children. What is the conditional probability that both are boys given that at least one of them is a boy? Assume that the sample space S is given by S = {(b,b),(b,g),(g,b),(g,g)}, and all outcomes are equally likely. ((b,g) means for instance that the older child is a boy and the younger child a girl.)
Also find the conditional probability that both are boys given that at least one of them is a girl?
Ans. 1/3, 0

Two people share a room in a hospital. On any given day a person independently picks up an infection with probability 1/4. An individual who contracts it on a day will definitely pass it on to the other person (in the room) the next day. Once contracted the infection stays for two days. What is the probability that at the end of 2 days both patients have the infection?
Ans. 121/256

[Amit Goyal]

21 Feb, 2009

A mine owner derives an income of Rs. 2000 this year and his income falls by Rs. 200 in each following year until no income results. Find the present value of the income stream when the interest is added yearly at
a) 4% p.a.
Ans. 2000+1800/1.04+1600/((1.04)(1.04))+.......

b) 5% p.a.
Ans. 2000+1800/1.05+1600/((1.05)(1.05))+.......

22 - 24 Feb, 2009

Show by means of numerical example that P(B|A)+P(B|A')
a) may be equal to 1;
Ans. Suppose a die is thrown. B is an event that 1 occurs. Let A = B, P(B|A)+P(B|A')=1

b) need not be equal to 1.
Ans. Suppose a die is thrown. B is an event that 1 occurs. Let A be the event that an odd number comes up, P(B|A)+P(B|A')=1/3

Given three events A, B and C such that P(A ∩ B ∩ C) ≠ 0, and P(C|A ∩ B) = P(C|B), show that P(A|B ∩ C) = P(A|B)
Ans. P(A|B ∩ C)=P(A ∩ B ∩ C)/P(B ∩ C)=P(A ∩ B ∩ C)P(A ∩ B)/P(B ∩ C)P(A ∩ B)
=P(C|A ∩ B)P(A ∩ B)/P(B ∩ C)=P(C|B)P(A ∩ B)/P(B ∩ C)=P(A ∩ B)/P(B)=P(A|B)

Show that P(A ∩ B ∩ C) = P(A).P(B).P(C) does not necessarily imply that A, B and C are all pairwise independent.
Ans. Suppose a die is thrown:
A = (1,2,3,4), P(A)=2/3
B = {1,2,3}, P(B)=1/2
C = {1,5,6}, P(C)=1/2
P(A).P(B).P(C) =1/6
A ∩ B ∩ C = {1}, P(A ∩ B ∩ C)=1/6
P(A ∩ B)= 1/2 ≠ 1/3 = P(A).P(B)
P(C ∩ B)= 1/6 ≠ 1/4 = P(C).P(B)
P(A ∩ C)= 1/6 ≠ 1/3 = P(A).P(C)


25-28 Feb, 2009
Income consumption curve for two goods x and y has equation y=2x. What will be the income elasticities for x and y?
Ans. Income elasticities of x and y equals 1.

A's and B's demand curves for apples are given by
p=20-q (A)
p=5-q/2 (B)
Suppose there are only two consumers in the market. Market Supply function is given by: p=2+Q.
Find the equilibrium quantity and price in the market.
Ans. Q=9, P=11

Laxmi is a poor agricultural worker. Her consumption basket comprises three commodities: rice and two vegetables - cabbage and potato. But there are occasionally very hard days when her income is so low that she can afford to buy only rice and no vegetables. However, there never arises a situation when she buys only vegetables and no rice. But when she can afford to buy vegetables, she buys only one vegetable, namely the one that has the lower price per kilogram on that day. Price of each vegetable fluctuates day to day while the price of rice is constant. Write down a suitable utility function that would represent Laxmi’s preference pattern. Explain your answer.
Ans. (p+c+1)(r-5), If m<5p(r), then laxmi will only buy rice otherwise she will buy p or c, which every is cheaper with r.

A person wants to sell his labor and spend his income entirely on the consumption of good G. His utility function is given by u= (1/2)log(G)+(3/4)log(L) where L is number of leisure hours. Maximum number of hours available in a day = 10 i.e. if he works for x hours then his leisure hours equal 10-x. Give the supply curve of labor. Assume that price of G=1.
Ans. He works for 4 hours regardless of the wage rate.

1st-4th Mar, 2009
WARP says that if a consumer chooses commodity bundle A when she can afford bundle B from any budget in which she can afford A. Check if WARP holds or not:
1) (x,y) = (2,3) and (p(x),p(y))=(1,4) ; (x’,y’) = (5,2) and (p(x’),p(y’))=(1,5)
2) (x,y) = (2,3) and (p(x),p(y))=(2,1) ; (x’,y’) = (3,2) and (p(x’),p(y’))=(1,2)
3) (x,y) = (2,3) and (p(x),p(y))=(1,2) ; (x’,y’) = (3,2) and (p(x’),p(y’))=(2,1)
Ans. 1) WARP holds;
2) WARP holds;
3) WARP does not hold.

Kamal has been given a free ticket to attend a classical music concert. If Kamal had to pay for the ticket, he would have paid up to Rs. 300/- to attend the concert. On the same evening, Kamal’s alternative entertainment option is a film music and dance event for which tickets are priced at Rs. 200/- each. Suppose also that Kamal is willing to pay up to Rs. X to attend the film music and dance event. What does Kamal do, i.e., does he attend the classical music concert, or does he attend the film music and dance show, or does he do neither? Justify your answer.
Ans. If X>500, attend a film music and dance event.
If X<500, attend a classical music concert.

Suppose there are two consumers A and B. Both consumers have a unit demand for the good. Consumer A has valuation $50 for it and consumer B's valuation is $20. What will be the monopolist price and quantity if he produces the good at 0 marginal cost? What will happen under duopoly i.e if there are two firms? What will be competitive output and quantity? Comment on DWL in three market structures.
Ans. Monopolist price = 50 and quantity = 1, DWL = 20
Cournot Duopolist price = 20 and each firm sells 1 unit each, DWL = 0
Competitive price = 0 and each firm sells 1 unit each, DWL = 0

You have 10 chocolates to divide among three individuals(A,B and C). All 3 prefer more chocolates to less.Find all the Pareto efficient allocations. Suppose individual A has 0 chocolates and the other two has 5 chocolates each. Find the competitive equilibrium price and allocation. Is the equilibrium pareto efficient?
Ans. Any feasible allocation is efficient, set is given by{x(a),x(b),x(c)|x(a)+x(b)+x(c)=10}
Competitive equilibrium is given by (0,5,5) with price of chocolate equal to 1.
It is efficient.