Question

12.3 (0) Clarence Bunsen is an expected utility maximizer. His pref- erences among contingent commodity bundles are represented by the ex- pected utility function

u(c1,c2,π1,π2)=π1√c1 +π2√c2.

Clarence’s friend, Hjalmer Ingqvist, has offered to bet him $1,000 on the outcome of the toss of a coin. That is, if the coin comes up heads, Clarence must pay Hjalmer $1,000 and if the coin comes up tails, Hjalmer must pay Clarence $1,000. The coin is a fair coin, so that the probability of heads and the probability of tails are both 1/2. If he doesn’t accept the bet, Clarence will have $10,000 with certainty. In the privacy of his car dealership office over at Bunsen Motors, Clarence is making his decision. (Clarence uses the pocket calculator that his son, Elmer, gave him last Christmas. You will find that it will be helpful for you to use a calculator too.) Let Event 1 be “coin comes up heads” and let Event 2 be “coin comes up tails.”

(a) If Clarence accepts the bet, then in Event 1, he will have _______

dollars and in Event 2, he will have ______________ dollars.
(b) Since the probability of each event is 1/2, Clarence’s expected utility

foragambleinwhichhegetsc1 inEvent1andc2 inEvent2canbe described by the formula ________________. Therefore Clarence’s

expected utility if he accepts the bet with Hjalmer will be ____________. (Use that calculator.)

(c) If Clarence decides not to bet, then in Event 1, he will have _________ dollars and in Event 2, he will have _______ dollars. Therefore if he doesn’t bet, his expected utility will be ________.

(d) Having calculated his expected utility if he bets and if he does not bet, Clarence determines which is higher and makes his decision accordingly.

Does Clarence take the bet? _____.

12.4 (0) It is a slow day at Bunsen Motors, so since he has his calcu- lator warmed up, Clarence Bunsen (whose preferences toward risk were described in the last problem) decides to study his expected utility func- tion more closely.

(a) Clarence first thinks about really big gambles. What if he bet his entire $10,000 on the toss of a coin, where he loses if heads and wins if tails? Then if the coin came up heads, he would have 0 dollars and if it came up tails, he would have $20,000. His expected utility if he took the

bet would be _____, while his expected utility if he didn’t take the bet would be _____. Therefore he concludes that he would not take

such a bet.

(b) Clarence then thinks, “Well, of course, I wouldn’t want to take a chance on losing all of my money on just an ordinary bet. But, what if somebody offered me a really good deal. Suppose I had a chance to bet where if a fair coin came up heads, I lost my $10,000, but if it came up tails, I would win $50,000. Would I take the bet? If I took the bet,

my expected utility would be ______. If I didn’t take the bet, my expectedutilitywouldbe ______.Therefore I should ____ the bet.”

(c) Clarence later asks himself, “If I make a bet where I lose my $10,000 if the coin comes up heads, what is the smallest amount that I would have to win in the event of tails in order to make the bet a good one for me to take?” After some trial and error, Clarence found the answer. You, too, might want to find the answer by trial and error, but it is easier to find the answer by solving an equation. On the left side of your equation, you would write down Clarence’s utility if he doesn’t bet. On the right side of the equation, you write down an expression for Clarence’s utility if he makes a bet such that he is left with zero consumption in Event 1 and x in Event 2. Solve this equation for x. The answer to Clarence’s

question is where x = 10,000. The equation that you should write is ______. The solution is x = __________.

(d) Your answer to the last part gives you two points on Clarence’s in- difference curve between the contingent commodities, money in Event 1 and money in Event 2. (Poor Clarence has never heard of indifference curves or contingent commodities, so you will have to work this part for him, while he heads over to the Chatterbox Cafe for morning coffee.) One of these points is where money in both events is $10,000. On the graph below, label this point A. The other is where money in Event 1 is zero

and money in Event 2 is ________. On the graph below, label this point B.

(e) You can quickly find a third point on this indifference curve. The coin is a fair coin, and Clarence cares whether heads or tails turn up only because that determines his prize. Therefore Clarence will be indifferent between two gambles that are the same except that the assignment of prizes to outcomes are reversed. In this example, Clarence will be indif- ferent between point B on the graph and a point in which he gets zero if

Event 2 happens and ________ if Event 1 happens. Find this point on the Figure above and label it C.

(f) Another gamble that is on the same indifference curve for Clarence

as not gambling at all is the gamble where he loses $5,000 if heads turn

up and where he wins _________ dollars if tails turn up. (Hint: To solve this problem, put the utility of not betting on the left side of an equation and on the right side of the equation, put the utility of having $10, 000 − $5, 000 in Event 1 and $10, 000 + x in Event 2. Then solve the resulting equation for x.) On the axes above, plot this point and label it D. Now sketch in the entire indifference curve through the points that you have labeled.

I want to know how to do this question with working. Please help me.

Answer 1

Answer below, (a) If Clarence accepts the bet, afterwards among outgrowth 1, that choice hold 9,000 bucks and among outgrowth 2, he will hold 11,000 dollars.
(b) Since the likelihood of each tournament is 1/2, Clarence’s anticipated assistance because a hazard of which she receives c1 of resultant 1 and c2 in outgrowth 2 be able be
described by using the formula
1 2√c1 + 1 2√c2. Therefore Clarence’s predicted good salvo he accepts the bet together with Hjalmer desire be 99.8746. (Use that calculator.)
(c) If Clarence decides not in imitation of bet, after of outgrowth 1, that wish hold 10,000 bucks then in outgrowth 2, that intention have 10,000 dollars. Therefore if that doesn’t bet, his expected necessity wish stand 100.
(d) Having deliberated his expected application condition that bets or if he does now not bet, Clarence determines who is higher then makes his selection accordingly. Does Clarence smoke the bet? No.