Economic Problem Set 2 Assignment Questions 6, 7 ,8 and 9

Problem Set 2Due November 186. Evaluateˆu?(w + irs) sF?(s) dsandˆu?? (w + irs) sF?(s) dsfor each of the utility functions above assume that s is distributed uniformlyon the interval [?1, 3] (which means that F (s) = s+14.7. A more interesting question than the one in the reading is to ask whetheran individual invests a higher proportion of his or her wealth in the riskyasset as their wealth rises. Use the method in the reading to prove thatthis is true provided the Arrow Pratt measure of relative risk aversionis decreasing in wealth where the Arrow Pratt measure of relative riskaversion is given by?u?? (w)u? (w)w.To do this, write the expected payoff asˆu ((1 ? ?) W + ?W (1 + z)) dF (z).Here ? is the proportion of wealth invested in the risky asset, so theoptimal proportion is derived by taking the derivative of this function andsetting it equal to zero. Differentiate this derivative implicitly as we didin class to findd? (W)dWthen mimic the argument we used in class to show that d?(W)dW is increasingprovided the investor’s relative risk aversion is declining in wealth.28. Ben loves hockey. He really enjoys it when his favorite team, the CalgaryFlames, are playing well. He has an income of y, and gets a utility ofu(y) when the Flames win. However, when the team loses, he gets upsetand says that he would give up $d dollars in order for the Flames to haveplayed better and won (so that his utility when the team loses is u(y?d)).Ben is risk averse, so that u??(·) < 0 (and u?(·) > 0).A betting agency offers the following deal: you can pay $q, and if theFlames lose you get a net payout of $b (i.e. b is in addition to gettingyour $q bet back). There is competition among betting agencies whichimplies that their profits are zero. Everyone (Ben, the betting agencies,and everyone else) evaluates the probability that the Flames will win at1 ? p.9. Write out the betting agency’s profit function, and write the zero-profitrelationship between q and b. How does q/b change as the Flames becomemore likely to win (i.e. as p decreases)?(a) Does Ben find it worthwhile to take the bet? That is, solve Ben’soptimal choice of b and q (subject to the zero-profit condition) anddetermine whether b? > 0.(b) Does the answer to the last part surprise you? In particular, howwould you reconcile the fact that Ben is betting on his favorite teamlosing? What about the fact that Ben is risk-averse?(c) What would p have to be in order for Ben to not make any bets(b? = 0)?3



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