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Decision Theory Under Certainty

(1) Decide whether the following statements are true or false and provide a brief explanation

a. The Expected Payoff of a lottery is defined as the sum of all the
possible payoffs.

b. An expected utility maximizer agent is indifferent between obtaining for sure (with probability one) the Certainty Equivalent of a lottery and the lottery.

c. The Risk Premium associated with a given lottery is the sum of the certainty equivalent and the expected payoff.

d. For a Risk Neutral agent the certainty equivalent of any lottery is just the expected payoff of the lottery.

e. For a Risk Averse agent the utility of obtaining for sure (with probability one) the expected payoff of the lottery is higher than the expected utility of the lottery.

f. The risk premium of any lottery is negative for a Risk Loving agent.

g. Diversifying, insuring and buying information are mechanisms to
reduce the risk of a lottery.

h. A risk averse agent must choose between the following two lotteries: (Lottery 1) $1,000 with probability 0.8 and $500 with probability 0.2. (Lottery 2) $900 for sure (with probability one). His optimal choice is Lottery 2.

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(2) Given the following utility function: 𝑢 𝑥 = 𝑥!, consider a lottery that pays $100 with probability 0.25 and $25 with probability 0.75.

a. Compute the certainty equivalent associated with the lottery.
b. Compute the risk premium associated with the lottery. What is
the interpretation of this magnitude?
c. Repeat (a) and (b) for the utility function: 𝑢 𝑥 = 𝑥.

(3) Assume that your initial wealth is $1,000. You can invest this money in buying an Apple stock or a government bond. The stock pays $1,000,000 if Apple obtains positive profits, $10,000 if profits are zero and you will lose all your money if the company bankrupts. The probability of bankruptcy is 0.1 and the probability of zero profits is 0.4. You can also buy Greek government debt which pays $90,000 with probability 0.8 and $625 with probability 0.2. If your utility function were u(x) = x1 2 , which is your optimal investment? What is the certainty equivalent associated
with the stock?

(4) Richard is the owner of a house with a market value of $9,000,000. A hurricane could damage his house and reduce the value to $4,000,000 with probability 0.1. An insurance company offers him an insurance policy which will cover completely the damage if he pays $500,000. Should Richard accept this insurance policy?

(5) Sebastian is a risk neutral entrepreneur who is the owner of a ranch which could contain oil. He knows that the ranch contains oil with probability 0.25. A company is willing to pay $90,000 for the ranch. If Sebastian decides to exploit the ranch he could obtain $700,000 if it contains oil and -$100,000 otherwise.

a. Should Sebastian exploit or sell the ranch?
b. An infallible expert is willing to analyze the ranch and tell
Sebastian if there is oil in the ranch. This expert is asking for
$30,000. Should Sebastian pay for this analysis?
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