# Which is largeró the expected utility of the lottery, or the utility of the expected value of the lottery?

This homework Örst examines the issues involved in using the expected utility representation of preferences to model choices over risky alternatives. Consider the expected utility function U(x1; x2) = 1 p x1 + 2 p x2; where x1 and x2 are (monetary) consumption levels in states 1 and 2, respectively, which occur with probabilities 1 and 2. Let 1 = 3 8 and 2 = 5 8 . (a) Calculate the expected utility of a lottery that pays nothing in state 1 and \$100 in state 2. Calculate the expected value of this lottery, and the utility of receiving this expected value with certainty, i.e., of receiving this expected value in both states of nature. Which is largeró the expected utility of the lottery, or the utility of the expected value of the lottery? What does this tell you about this personís attitude toward risk? (b) The certainty equivalent of the lottery is a payo§ which, if received with certainty, would make the person indi§erent between the lottery and receiving the certainty equivalent in each state. Calculate the certainty equivalent of the lottery in [a]. How does it compare to the expected value? (c) Show that, in general, if one is risk averse, then the certainty equivalent of a lottery falls short of its expected value. The di§erence between these two is often called the risk premium for the lottery. (d) Find the formula for an indi§erence curve, giving x2 as a function of x1, identifying combinations of x1 and x2 that give the same expected utility. (e) Find the slope of this indi§erence curve when x1 = x2. Explain why it has this slope. (f) Show that the preferences described by this expected utility function are convex

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