3. This question examines the issues involved in using the expected utility representation of preferences to model choices over risky alternatives. Consider two lotteries. Lottery 1 gives payoff 0 with probability 1/2 and payoff 20 with probability 1/2. Lottery 2 gives payoff 0 with probability 49/50 and payoff 500 with probability 1/50. (a) Calculate the expected payoff of each lottery. (b) Suppose an agent has the expected utility function U(X1, 12) = mu(x1) + 2u(12), where u(r;) = 12. Show that this person is risk seeking. Calculate their expected utility under the two lotteries. Which do they prefer? (c) Find the certainty equivalent for each of the two lotteries, i.e., for each lottery, find the amount of money which, if received with certainty, would make an individual indifferent between the lottery and the certain amount of money. (d) Suppose now that each of these lotteries was offered for sale at the certainty equivalent you have calculated in (c). The seller collects the sale price of the lottery as income, and then pays out the realized value of the lottery. This might be the case if lottery buyers had underlying utility functions u(r;) = r,, and sellers priced the lotteries at the highest value buyers where willing to pay, i.e., the certainty equivalent. Suppose sellers are risk neutral. What is the seller's expected payoff from selling each lottery? Which lottery is more lucrative for them? (e) Would you expect the typical patron betting on horse races to be risk averse, risk neutral, or risk seeking? In light of this and your answers to (a) (d), do you think a risk-neutral person would be better off betting on favorites (lotteries featuring relatively modest payoffs with relatively high probability) or long shots (lotteries with small probabilities of large payoffs)