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suggested exercise set 02 solutions

# suggested exercise set 02 solutions - Econ 110 Pol Sci 135...

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Econ 110 / Pol Sci 135 Suggested Exercises Set 2 Solutions Fall 2009 Page 2 of 10 2. The table of certainty equivalents is as follows: (a) In order to plot the utility function, we need to show the relationship between a given input, the dollar value, which we can put on the x-axis, and the utility of that dollar value, which we put on the y-axis. Note that because the utility of \$1000 is 10 and the probabilities above denote probabilities of winning \$1000, we have, for example 0.8* u (1000)= u (512). u (1000)=10, so 8= u (512). We can graph this system: 0 1 2 3 4 5 6 7 8 9 10 0 100 200 300 400 500 600 700 800 900 1000 Dollar Payment Utility of Dollar Payment
Econ 110 / Pol Sci 135 Suggested Exercises Set 2 Solutions Fall 2009 Page 3 of 10 Note that the utility relationship is the familiar cube-root utility that we have seen in lecture. So, the utility function is: 0 1 2 3 4 5 6 7 8 9 10 0 45 90 135 180 225 270 315 360 405 450 495 540 585 630 675 720 765 810 855 900 945 990 Dollar Payment Utility of Dollar Payment (b) The expected utility of the lottery is ( ) [ ] ) u(outcome2 * 2} Pr{outcome ) u(outcome1 * 1} Pr{outcome + = L U E . In this case, the expected utility is ( ) [ ] ( ) ( ) 2.0127 2.1544 4 3 1.5874 4 1 10 4 3 4 4 1 10 4 3 4 4 1 3 1 3 1 = + = + = + = u u L U E The certainty equivalent of the lottery is the dollar value that will yield a utility equal to the utility of the lottery. So, u(Certainty Equivalent) = 2.0127. So, 0127 . 2 3 1 = CE Which implies 1533 . 8 0127 . 2 3 = = CE . (c) Using the reasoning from part (b), ( ) [ ] ( ) ( ) ( ) ( ) 3 1 3 1 3 1 10 24 7 6 4 1 2 3 1 0 8 1 10 24 7 6 4 1 2 3 1 0 8 1 + + + = + + = u u u u L U E Solving, ( ) [ ] 5477 . 1 6824 . 0 4453 . 0 4200 . 0 2144 . 2 24 7 8171 . 1 4 1 2599 . 1 3 1 0 = + + = + + + = L U E

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Econ 110 / Pol Sci 135 Suggested Exercises Set 2 Solutions Fall 2009 Page 4 of 10 3. (a) There is no pure-strategy Nash equilibrium here, hence the search for an equilibrium in mixed strategies. Row’s p-mix (probability p on Up) must keep Column indifferent and so must satisfy 16p + 20(1 – p) = 6p + 40(1 – p); this yields p = 2/3 = 0.67 and (1 – p) = 0.33.
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