Question

1. Suppose a fair, two-sided coin is flipped. If it comes up heads you receive $5. If it comes up tails you lose $1. The expected value of this lottery is (a) $2 (b) $3 (c) $4 (d) $5 (e) None of the above

2. An individual has a vNM utility function over money of u(x) = p3 x , where x is final wealth. She currently has $8 and can choose among the following three lotteries. Which lottery will she choose? • Lottery 1: Give up her $8 and face the gamble (0.1, 0.5, 0.4) over final wealth levels ($1, $8, $27). • Lottery 2: Keep her $8. • Lottery 3: Give up her $8 and face the gamble (0.2, 0.8,0.0) over final wealth levels ($1, $8, $27) (a) Lottery 1 (b) Lottery 2 (c) Lottery 3 (d) She is indifferent between the three lotteries.

3. An individual has a vNM utility function over money of u(x) = px, where x is final wealth. Assume the individual currently has $16. He is offered a lottery with three possible outcomes; he could gain an extra $9, lose $7, or not lose or gain anything. There is a 15% probability that he will win the extra $9. What probability, p, of losing $7 would make the individual indifferent between to play and to not play the lottery? (a) p = 0.15 (b) p = 1.08 (c) p = 0.415 (d) p = 0.05 (e) None of the above

Answer 1

1. EXPECTED VALUE OF LOTTERY = P(HEAD).$5 + P(TAIL)(-)$1 = 0.5*5-0.5*1 = 0.5*4 = 2$ (A) IS CORRECT

2. U = P3.X. FINDING EXPECTED UTILITIES.

prob of 1st payoff. U[1st payoff] + prob of 2nd. U[2nd payoff] + prob of 3rd payoff.U[3rd payoff]

1ST LOTTERY EU = 0.1*(1-8)+0.5*(8-8)+0.4*(27-8) = -0.1*7 +0.4*19 = 0.06

2nd lottery, EU = 0.2*(1+8)+0.8*(8+8)+0.0*(27+8) = 0.2*9 + 0.8*16 = 2.28

3RD LOTTERY, EU = 0.2*(1-8)+0.8*(8-8)+0.0*(27-8) = -0.2*7 = -1.4

HE WILL CHOOSE THE 2ND LOTTERY WITH MAXIMUM EXPECTED UTILITY.

Q3. EXPECTED UTILITY FROM NOT PLAYING = P.16. SINCE P=1 AS THE GAME IS CERTAIN, EU = 16

IF HE CHOOSES TO PLAY THE GAMBLE, EU =0.15*(16+9) + P'*(16-7) + P''(16) = 3.75 + 7P'+16P''

WHERE P' is the probability of losing 7$ and P'' is prob of not gaining or losing anything.  he will be indifferent between playing and not playing if his EU from two outcomes is same

3.75 + 7P' +16P'' = 16

7P' + 16P'' =12.25

since we donot know P'', we cannot find P'