1.8 Bibliographical References, Extensions and Exercises (1.3) Let u w2 for w o.

Question

1.8 Bibliographical References, Extensions and Exercises (1.3) Let u w2 for w o. (a) Compute the exact risk premium if initial wealth is 4 and if a decision maker faces the lottery (-2, 2; +2, Explain why the risk premium is negative. If the utility function becomes v w4, what happens to the risk pre (b) Show that v is a convex transformation of u. mium? In w (1.4) Let u (a) Does this utility function exhibit the DARA property? (b) Compute -u" /u" and compare it with -u theorem). (1.5) consider the family of exponential utility functions l exp(-aw) (a) Show that a is the degree of absolute risk aversion. (b) show that u becomes linear in w when a tends to zero Chint: use L'Hopital's rule). (c) Consider lottery with positive and negative payoffs. Determine the value of Eu() when a tends to infinity.

Explanation / Answer

1.3 a) If the individual chooses to play the lottery then his expected utility E(u) = 0.5 (4+2)2 + 0.5 (4-2)2 = 20

Now, Let the amount required to make the individual indifferent between playing and not playing the lottery be x

Then, Certainty equivalent x becomes (4+ x)2 = 20 or x = 4.4721 - 4 = 0.4721

So, Risk premium = - 0.4721

Now, for this individual U= w2 so this guy is risk loving as U' = 2w and U'' = 2 and r(w) = U'' / U' = 1/w, thus as the wealth increases his risk aversion decreases and he wants to play the lottery there is no need for anyone to pay him a positive premium to play the lottery instead you pay him a negative premium

Another useful analogy would be supposingly you are at a party and you have two friends A & B. Now, A is an alcholic and B had never done alchol. You want to have fun and see how B behaves when he is high. Now, you would go an extra mile and pay B some premium so that he tries alchol and you can see what happens to him whereas everybody knows when A drinks he creates a scene at the party so everybody would be charging him a fee to tolerate his nuisance, this fee is like a negative premium

b) Now, if utility function becomes v = w4 then expected utility becomes 656 and the certainity equivalent is 1.06 and the risk premium is -1.06, the risk premium increases because this guy is even more risk loving as v' = 4w3 and v'' = 12w2 so r(w) = 3/w > 1/w

Now, v= w4 and u = w2 or v = u2 thus v is a convex transformation of u

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