An individual with utility function U(X) = X2 bought Lottery A and Lottery B.
Lottery A gives a payoff of $2 or $4 equally likely.
Lottery B gives a payoff of $4 or $8 equally likely.
a) Calculate the expected utility of Lottery A and Lottery B together.
b) What is the certainty equivalent of Lottery A and Lottery B together?
c) Find the maximum price which this individual is willing to pay for both lotteries.
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To calculate the expected utility, we need to determine the utility associated with each possible outcome and then calculate the weighted average of the utilities based on the probabilities.
a) For Lottery A:
The payoff of $2 has a probability of 0.5, and the payoff of $4 also has a probability of 0.5. So the expected utility of Lottery A can be calculated as follows:
Expected utility of Lottery A = (U($2) * Probability of $2) + (U($4) * Probability of $4)
Since the utility function is U(X) = X^2, the expected utility of Lottery A is:
Expected utility of Lottery A = (2^2 * 0.5) + (4^2 * 0.5)
= (4 * 0.5) + (16 * 0.5)
= 2 + 8
= 10
For Lottery B:
The payoff of $4 has a probability of 0.5, and the payoff of $8 also has a probability of 0.5. So the expected utility of Lottery B can be calculated as follows:
Expected utility of Lottery B = (U($4) * Probability of $4) + (U($8) * Probability of $8)
Again, using the utility function U(X) = X^2, the expected utility of Lottery B is:
Expected utility of Lottery B = (4^2 * 0.5) + (8^2 * 0.5)
= (16 * 0.5) + (64 * 0.5)
= 8 + 32
= 40
b) The certainty equivalent is the guaranteed amount of money that the individual would be willing to accept instead of participating in the lottery. To find the certainty equivalent, we need to find the amount of money that gives the individual the same utility as the expected utility of the lottery.
For Lottery A:
We need to find the amount of money, let's call it x, for which U(x) is equal to the expected utility of Lottery A, which is 10. In this case, the equation is:
x^2 = 10
Solving this equation, we find that x ≈ 3.16. So the certainty equivalent for Lottery A is approximately $3.16.
For Lottery B:
Using the same logic, we need to find the amount of money, let's call it y, for which U(y) is equal to the expected utility of Lottery B, which is 40. In this case, the equation is:
y^2 = 40
Solving this equation, we find that y ≈ 6.32. So the certainty equivalent for Lottery B is approximately $6.32.
c) To find the maximum price that the individual is willing to pay for both lotteries, we need to calculate the expected utility of paying for the lotteries and subtract the certainty equivalents from that expected utility.
Expected utility of paying for both lotteries = (Expected utility of Lottery A) + (Expected utility of Lottery B)
= 10 + 40
= 50
Now, we subtract the certainty equivalents from the expected utility:
Maximum price = Expected utility of paying for both lotteries - (Certainty equivalent for Lottery A + Certainty equivalent for Lottery B)
= 50 - (3.16 + 6.32)
= 50 - 9.48
= 40.52
Therefore, the maximum price that the individual is willing to pay for both lotteries is approximately $40.52.