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Here were the questions for the utility measurement part, in the order in which they are in the data. (Some got them in reverse order.)
Difference question
Which would have a greater effect on what is important to you about your life at age 30? * The difference betwen an annual (pre-tax) income of $40,000 and $X, or * The difference between an annual (pre-tax) income of $X and $120,000.
Here enter (by cutting and pasting or retyping) the single line for your ten difference judgments from the SECOND email you received with your results from the first part: If you did not do the first part in time, you can still answer most of the questions here, so you can get some credit for this assignment; just leave this blank.
Gamble question Supposed you had a choice of two jobs. Each would provide your sole income at age 30. Which would you choose? * This one would pay $X per year (in current dollars). * This one has a 50% chance of paying $40,000 per year and a 50% chance of paying $120,000.
Here enter the single line for your gamble judgments:
Explanation of data for Gamble and Difference questions
The values of X were 50,000, 60,000, 70,000, 80,000, and 90,000. The sequence was repeated with everything doubled, so that, for example, the range was $80,000-$240,000 instead of $40,000-$120,000. This was mainly just to get more data. Note that the midpoint of the numbers (70,000) is not the middle of the range you were given (40,000 to 120,000), which is 80,000.
In the data, 1 represents a choice that implies a less concave utility function and 0 represents a choice that implies a more concave utility function. That is, 1 represents a choice of the riskier investment, or a judgment that the higher interval is larger than the lower one (e.g., the difference between $X and $120,000 is greater than the difference between $40,000 and $X). For each subject, the score was simply the mean judgment. For example, if the judgments were "1 1 1 0 0 1 1 1 0 0" the score would be .6. Note that $80,000 (not $70,000) is the midpoint between $40,000 and $120,000.
Questions about Gamble and Difference
If a subject got a score of 1.0, what would that imply about the utility function; is it convex, concave, or linear?
What scores would be consistent with a linear utility function (no concavity and no convexity)? Note that one of the items in each group of 5 would yield indifference in this case, but you were forced to answer, so there could more than one score consistent with linear utility.
Note that this scoring method assumes that subjects were consistent. For example, a rejection of a gamble for a given value of X should imply rejection of the gamble when the sure thing (X) is even higher. Some responses were not consistent, but I just assumed that these were random errors and the mean response is still the best estimate we could get.
Were all your responses consistent? If not, was the incosistency in the gambles, the differences, or both? Make sure to look within each group of 5. You do not need to be consistent between the two groups of 5.
Here are summary statistics for the two measures of the utility function:
Gambles Min. 1st Qu. Median Mean 3rd Qu. Max. 0.1000 0.3000 0.4000 0.4505 0.6000 1.0000 Differences Min. 1st Qu. Median Mean 3rd Qu. Max. 0.0000 0.4000 0.6000 0.5645 0.8000 1.0000
Are utility functions inferred from your risk choices more concave (more risk averse) than those inferred from your difference judgments?
Luxury questions
The first 33 questions were about your taste for luxury. After each item, you were asked: How does this affect what is important to you about your life as a retiree or senior? 0 I don't care about this at all. 1 This would be nice, but it would have little effect. 2 This would have a noticeable effect. 3 This would have a large effect. 4 This is absolutely essential.
The most essential items were (most essential on top): * having a high-speed (fiber optic or cable) Internet connection * saving $50,000/year, as opposed to saving nothing * saving $100,000/year, as opposed to saving $50,000 * $1,000/year for clothing (including accessories and jewelry, beyond minimal maintenance * flying to see relatives (including children) or friends once a year
The least essential, hence most luxurious, items were: * owning a second $20,000 car (as opposed to one $20,000 car) * $20,000 for a second extra bedroom in your home for visitors * $3,000/year for regularly (twice per week) buying wine that costs about $30/bottle, as opposed to $1,000/year for wine that costs $10/bottle * $5,000/year for someone to maintain a garden or lawn * $1,000/year for regularly (twice per week) buying wine that costs about $10/bottle
To assess the taste for luxury, I computed a measure called Lux. For each subject, I found the slope of the best-fitting regression line for predicting that subject's responses to these items from the mean responses of all subjects across the 34 items. The following two plots show the two members of the class with the highest and lowest slopes. Each point is an item. The vertical axis is the subject's response to that item. The horizontal axis is the class mean for it. The lines are the best fitting predictors of the subject's ratings from the class mean ratings of the 34 items.
The idea here is that a high slope means that the subject cares relatively little for the things that most others care about less; these are the luxuries. A low slope means that the subject cares just as much about these things as the things that other subjects regard as essential. Thus, a high slope means LESS of a taste for luxuries. Those with a high slope should have a more concave utility function. It would be more important for such people to have a safe but basic income rather than taking a risk in hopes of hitting the jackpot, because they would not benefit that much from the extra income.
Here is the summary for Lux:
Min. 1st Qu. Median Mean 3rd Qu. Max. 0.189 0.796 1.021 1.000 1.266 1.669
Of interest is whether Lux correlated with the utility measures, Gambles and Differences. It did not (r=-.09 for gambles, .09 for differences, neither close to significant). Interestingly, it did correlate with gender: females 1.13, males 0.92 (t92=3.36, p=.001).
However, using a different situation, I have found that Differences do correlate negatively (as predicted) with Lux, and Gambles do not correlate at all. (And the two measures were significantly different.)
What does this difference (which we did not find here) imply about gambles and differences in terms of which is a better measure of predicted utility?
Why isn't Lux a perfect measure of the concavity of the utility function?
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