. Consider each of the following choices between a "sure thing" and a gamble. In the first situation, for example, imagine a choice between getting $0 for sure and a gamble in you could win $20 or lose $12 (I've written the amount as -$12 in the table). In the situations below, you can't actually make a choice, because I have not specified the probabilities of winning and losing in the gambles. That's your job. For each of the lity of winning P(Win) would have to be for you to be completely inquirerent to indifferent (or completely torn) between the sure thing and the gamble. There are only two possibilities in each gamble, so P(Lose) = 1 -P(Win). Fill in the table with your values of P(Win). Probability of Valve tha Option A Option B (Gamble) Indifference ) Utility Sure Thing S to Win S to Los P ( Win ) Us ) So - 2. Pin down the utility scale by assuming that U($0) = 0 and U($20) - 20. Except for the order ($20 is better than $0), these utilities are arbitrary. In principle, you could choose other utilities, like 0 and 1 or even -17 and 124. But 0 and 20 is a convenient pair. 3. In each situation, use algebra to compute the utility of the missing amount (the amount that is not 50 or $20). This is the "standard gamble" method for cal hod for calculating utilities. For in the first situation, if I said tha on, if I said that P(Win) = 0.6 would make me indifferent between a sure $0 and a gamble between $20 and -$12, I'd set the expected utilities of the two options equal (that's what it means to be indifferent in these situations) and solve for U(-$12). U($0) = 0.6 x U($20) + 0.4 x U(-$12) 0 = 0.6 x 20 + 0.4 x U(-$12) -12504 x 06312) -12/0.4 = U(-$12) -30 = U(-$12) So I'd write -30 in the first blank if the rightmost column in the table. Of course, I probably wouldn't write out the algebra in as much detail as I did above (that was just for clarity). In some cases, the algebra will be a bit easier. . Make a scatter plot of your utilities as a function of the dollar values. In other words, put the fatal axis and the calculated utilities on the vertical cats. Based on the above calculation, for example, I'd plot the point x = -$12y- -50. There should be 22 points in your plot: one for each of the situations in the table, plus the two points that were assumed at the beginning: (x = $0, y = 0) and (x = $20, y = 20). Your plot is likely to be very noisy. That often happens with data from an individual person, unless he or she adjusts the values for P(Win) to be more consistent (which is definitely okay, as consistency is a good thing). A hand-drawn plot is fine. . Your utility function is the best fitting line or curve through your scatter plot. It should probably go through the point (x = $0, y = 0). Comment on the shape of your utility function. Specifically, is your utility function any steeper for losses than for gains? Is it linear or is it curved? Is the curvature for losses different from the curvature for gains? . OPTIONAL. You may want to create your plot using the posted spreadsheet. You'll just input your P(Win) values and the utilities and plot will be created for you, which means that you don't have to do the algebra yourself. But definitely try the algebra for yourself to make sure you know how to do it. If the spreadsheet is too much of a hassle, then don't bother. WHAT TO TURN IN. Turn in your completed table from Part 1, the scatter plot of your utilities from Part 4 or 6 (either is fine), and the answers to the questions in Part 5. There is nothing to turn in from Part 2, and you don't need to hand in your algebra from Part 3. . Imagine that you would like to know how risk averse your friend is. Your friend tells you that her utility function is U(x) = x" and also that her a value is either 0.6 or 0.8. To help figure out which it is, you give her five choice problems and record her answers. Here is the data that you collect on her: she chooses $4 over a 50/50 gamble between $0 and $10. She chooses $13 over a 50/50 gamble between $0 and $28. She chooses $5 over a 50/50 gamble between $0 and $14. She chooses a 50/50 gamble between $0 and $9 over $2. She chooses a gamble with a probability p=0.2 of $20 and p=0.8 of $0 over $3. (a) For each of these choice problems there is a value a = a* that would make a person indifferent between the two options. That is, for some value a*, the utility of one option is equal to the utility of the other option. For each of the five choice problems, calculate this ndifference value a*. Note: Using the logarithm will help you here. (b) Take a given choice problem with indifference value a*. If your friend chooses the "sure" option on that choice problem, this suggests that her true a value is lower than a* (because she is relatively risk averse). On the other hand, if she chooses the "gamble" option on that choice problem, this suggests that her true a value is higher than a* (because she is relatively risk seeking). Based purely on the number of choices that ar choices that are consistent with the more likely that s true a = 0.6 or a = 0.8? Explain your answer. Instructions Convenient Assin A calculated utility function WeChat IX His