Question

It is a slow day at Bunsen Motors, so since he has his calculatorwarmed up, Clarence Bunsen decides to study his expected utility functionmore closely.

a)Clarence first thinks about really big gambles. What if he bet hisentire $10,000 on the toss of a coin, where he loses if heads and wins iftails? Then if the coin came up heads, he would have 0 dollars and if itcame up tails, he would have $20,000.His expected utility if he took thebet would be_________________________________, while his expected utility if he didn’t take thebet would be ________________________.Therefore he concludes that he would not takesuch a bet.

b)Clarence then thinks, “Well, of course, I wouldn’t want to take achance on losing all of my money on just an ordinary bet. But, whatif somebody offered me a really good deal. Suppose I had a chance tobet where if a fair coin came up heads, I lost my $10,000, but if it cameup 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, myexpected utility would be _____________________________.Therefore I should __________________ the bet.”

c)Clarence later asks himself, “If I make a bet where I lose my $10,000if the coin comes up heads, what is the smallest amount that I would haveto win in the event of tails in order to make the bet a good one for meto 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 tofind 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 rightside of the equation, you write down an expression for Clarence’s utilityif he makes a bet such that he is left with zero consumption in Event 1and x in Event 2. Solve this equation for x. The answer to Clarence’squestion is where x = 10, 000. The equation that you should write is________________________________________.The solution is __________________________________________________________

d)Your answer to the last part gives you two points on Clarence’s indifferencecurve between the contingent commodities, money in Event 1and money in Event 2. Oneof these points is where money in both events is $10,000. On the graph, label this point A. The other is where money in Event 1 is zeroand money in Event 2 is ______________. On the graph, label thispoint B.

e)You can quickly find a third point on this indifference curve. Thecoin is a fair coin, and Clarence cares whether heads or tails turn up onlybecause that determines his prize. Therefore Clarence will be indifferentbetween two gambles that are the same except that the assignment ofprizes to outcomes are reversed. In this example, Clarence will be indifferentbetween point B on the graph and a point in which he gets zero if Event 2 happens and _____________________if Event 1 happens. Find this pointon the graphand label it C.

Answer #1

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

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