Question

I need the working for the answers please.

Clarence Bunsen is an expected utility maximizer. His preferences among contingent commodity bundles are represented by the expected utility function u(c1,c2,?1,?2)=?1?c1 +?2?c2.

It is a slow day at Bunsen Motors, so since he has his calculator warmed up, Clarence Bunsen (whose preferences toward risk were described in the last problem) decides to study his expected utility function 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 expected utility would be ________. 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 indifferent 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.

Answer 1

Please find the answer below. Please note that I have taken the utility fucntion as (C₁^.5)/2+(C₂^.5)/2. I am assuming that the 1/2 are the part of utility function and not the probabilities attached to the utility function. The confusion arises because the probabilities in a fair coin toss are also 1/2 for each. In case these are indeed probabilities attached to the utility function, the calculations would have to be modified accordingly.