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Let Z = R+. Consider the following lotteries:

L I 1 = [$100]

L I 2 = 0.75 [$150] 0.25 [$1]

L II 1 = 1/3 [$100] 2/3 [$1]

L II 2 = 0.25 [$150] 0.75 [$1]

(a) Depict each of the lotteries L I 1 , L I 2 , L II 1 , and L II 2 in a tree diagram.

(b) Let L = 1/2 L I 1 1/2 L I 2 . Depict L in a tree diagram (keep L compound; do not reduce it to a simple lottery over Z).

(c) Find the simple lottery corresponding to the compound lottery L from part (b) and depict it in a tree diagram.

(d) Find the expected value of each of the lotteries L I 1 , L I 2 , L II 1 , L II 2 , and L.

(e) Bob is an expected utility maximizer who strictly prefers L I 1 to the lottery 0.5 [$0] 0.5 [$200]. Provide an example of a (Bernoulli) utility function u : Z R that is consistent with his preferences. Explain. In parts (f)(i), let Bobs preferences be represented by the utility function you provided in part (e).

(f) Compute Bobs expected utility for each of the lotteries L I 1 , L I 2 , L II 1 , and L II 2 .

(g) Which lottery would Bob choose out of each of the following sets: S I = {L I 1 , LI 2} and S II = { L II 1 , LII 2} ?

(h) Compute the coefficients of absolute and relative risk aversion for Bob. Are Bobs preferences IARA/ CARA/ DARA? Are they IRRA/ CRRA/ DRRA?

(i) Give another utility function, u, that also represents Bobs preferences.

(j) Jen chooses L I 1 out of S I and L II 2 out of S II (where the sets are as defined in part (g)). Both choices are strict, i.e., she strictly prefers the item chosen to the one not chosen. Prove directly (without referring to any theorems) that Jen is not an expected utility maximizer. (Hint: Use proof by contradiction.)

(k) 15 marks Show that Jens preferences violate the Independence axiom. (Hint: To show this, you must define the lotteries L1, L2 and L3 from the axiom and then express the lotteries from S I and S II in terms of L1, L2, and L3 to show how preferences between L1 and L2 imply preferences over S I and S II .)

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