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Re: Daily Question solutions

Posted by Amit Goyal on Mar 08, 2009; 2:21pm
URL: http://discussion-forum.276.s1.nabble.com/Daily-Question-solutions-tp2382507p2444626.html

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.